Algebra and Geometry of Elementary Functions
2021-01-27
Introduction
This notebook is intended to give a brief introduction to elementary functions emphasizing on effective thinking in algebra and geometry.
In the first part, we will review mathematical operations including addition, multiplication, -th root, exponentiation and logarithm.
In the second part, we will study the concepts of functions, algebraic functions and their applications.
In the third part, we will study elementary transcendental functions and applications.
Comments and suggestions are very welcome.
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
Part I: Mathematical Operations
Integer Exponents
Don’t Be Tricked
A pizza shop sales 12-inches pizza and 8-inches pizza at the price $12/each and $6/each respectively. With $12, would you like to order one 12-inches and two 8-inches. Why?
A sheet of everyday copy paper is about 0.01 millimeter thick. Repeat folding along a different side 20 times. Now, how thick do you think the folded paper is?
Properties of Exponents
For an integer , and an expression
, the mathematical operation of the
-times repeated multiplication of
is call exponentiation, written as
, that is,
In the notation ,
is called the exponent,
is called the base, and
is called the power, read as “
raised to the
-th power”, “
to the
-th power”, “
to the
-th”, “
to the power of
” or “
to the
”.
Property | Example |
---|---|
The product rule | |
The quotient rule (for | |
The zero exponent rule (for | |
The negative exponent rule (for | |
The power to a power rule | |
The product raised to a power rule | |
The quotient raised to a power rule (for |
Order of Basic Mathematical Operations
In mathematics, the order of operations reflects conventions about which procedure should be performed first. There are four levels (from the highest to the lowest):
Parenthesis; Exponentiation; Multiplication and Division; Addition and Subtraction.
Within the same level, the convention is to perform from the left to the right.
Example: Simplify. Write with positive exponents.
Solution. The idea is to simplify the base first and rewrite using positive exponents only.
Simplify (at least partially) the problem first
To avoid mistakes when working with negative exponents, it’s better to apply the negative exponent rule to change negative exponents to positive exponents and simplify the base first.
Practice
Exercise: Simplify. Write with positive exponents.
Exercise: Simplify. Write with positive exponents.
Exercise: Simplify. Write with positive exponents.
Exercise: Simplify. Write with positive exponents.
Exercise: A store has large size and small size watermelons. A large one cost $4 and a small one $1. Putting on the same table, a smaller watermelons has only half the height of the larger one. Given $4, will you buy a large watermelon or 4 smaller ones? Why?
Review of Factoring
Can You Beat a Calculator
Do you know a faster way to find the values?
- Find the value of the polynomial
when
.
- Find the value of the polynomial
when
.
- Find the value of the polynomial
when
.
- Find the value of
.
Factor by Removing the GCF
The greatest common factor (GCF) of two terms is a polynomial with the greatest coefficient and of the highest possible degree that divides each term.
To factor a polynomial is to express the polynomial as a product of polynomials of lower degrees. The first and the easiest step is to factor out the GCF of all terms.
Example: Factor .
Solution.
- Find the GCF of all terms.
The GCF of,
and
is
.
- Write each term as the product of the GCF and the remaining factor.
,
, and
.
- Factor out the GCF from each term.
.
Factor by Grouping
For a four-term polynomial, in general, we will group them into two groups and factor out the GCF for each group and then factor further.
Example: Factor .
Solution.
For a polynomial with four terms, one can normally try the grouping method.
- Group the first two terms and the last two terms.
- Factor out the GCF from each group.
- Factor out the binomial GCF.
Example: Factor .
Solution.
- Group the first term with the third term and group the second term with the last term.
- Factor out the GCF from each group.
- Factor out the binomial GCF.
Guess and check.
Once you factored one group, you may expect that the other group has the same binomial factor so that factoring may be continued.
Factor Difference of Powers
Factoring is closely related to solving polynomial equations. If a polynomial equation has a solution
, then
has a factor
. For example,
has a solution
. So the difference
has a factor
. Using long division or by induction, we obtain the following equality.
Difference of -th powers
In particular,
Example: Factor .
Solution.
- Recognize the binomial as a difference of squares.
- Apply the formula.
Example: Factor completely.
Solution.
Factor Trinomials
If a trinomial ,
, can be factored, then it can be expressed as a product of two binomials:
By simplify the product using the FOIL method and comparing coefficients, we observe that
A trinomial is also called a quadratic polynomial. The function defined by
is called a quadratic function.
Trial and error.
The observation suggests to use trial and error to find the undetermined coefficients ,
,
, and
from factors of
and
such that the sum of cross products
is
. A diagram as shown in the following examples will be helpful to check a trial.
Example: Factor .
Solution. One may factor the trinomial in the following way.
Factor
:
Factor
:
Choose a proper combination of pairs of factors and check if the sum of cross product equals
:
This step can be checked easily using the following diagram.Factor the trinomial
Example: Factor .
Solution. One may factor the trinomial in the following way.
Factor
:
Factor
:
Choose a proper combination of pairs of factors and if the sum of cross products equals
:
This step can be checked easily using the following diagram.Factor the trinomial
Use Auxiliary Problem.
Some higher degree polynomials may be rewrite as a trinomial after a substitution. Factoring the trinomial helps factor the polynomial.
Example: Factor the trinomial completely.
Solution. One idea is to use a substitute.
- Let
. Then
.
- Factor the trinomial in
:
.
- Replace
by
and factor further.
Practice
Exercise: Factor out the GCF.
Exercise: Factor by grouping.
Exercise: Factor completely.
Exercise: Factor completely.
Exercise: Factor the trinomial.
Exercise: Factor the trinomial.
Exercise: Factor completely into polynomials with integer coefficients.
Exercise: Each of trinomial below has a factor in the table. Match the letter on the left of a factor with a the number on the left a trinomial to decipher the following quotation.
“,
;
,
;
,
.”
A: | B: | C: | D: | E: | F: | G: |
H: | I: | J: | K: | L: | M: | N: |
O: | P: | Q: | R: | S: | T: | U: |
V: | W: | X: | Y: | Z: |
Algebra of Rational Expressions
Is There Enough Time
Matt is kayaking upstream on a river with his best effort. After 30 minutes, he received an emergency call and has to return in 20 minute. The speed of the current of the river is 1 mph. Under normal condition, a paddler’s average paddling speed is between 2 and 5 mph. Do you think Matt can return on time? Why?
A construction team is building a house. After half of the work was done, to expedite the construction process, the another team joins in the construction. The first team normally takes 7-10 days to build a hours. The second team normally takes 2 extra days to build a house. How many days it takes to build the house?
Rational Expressions
Let and
be polynomial functions of
and
is not a constant function. We call the function
a rational function. The domain of
is
. The expression
is called a rational expression, the polynomial
is called the numerator, and the polynomial
is called the denominator. A rational expression is simplified if the numerator and the denominator have no common factor other than
.
Let ,
be polynomials with
and
be a nonzero expression. Then
Example: Simplify .
Solution.
Factor both the top and the bottom.
Divide out common factors.
Example: Simplify .
Solution.
Factor both the top and the bottom.
Divide out common factors.
Multiplying Rational Expressions
If ,
,
,
are polynomials with
and
, then
Example: Multiply and then simplify.
Solution.
Factor numerators and denominators.
Multiply and simplify.
Example: Multiply and then simplify.
Solution.
Dividing Rational Expressions
If ,
,
,
are polynomials where
,
and
, then
Example: Divide and then simplify.
Solution.
- Rewrite as a multiplication.
- Factor and simplify.
Adding or Subtracting Rational Expressions with the Same Denominator
If ,
and
are polynomials with
, then
Example: Add and simplify
Solution.
- Determine if the rational expressions have the same denominator. If so, the new numerator will be the sum/difference of the numerators.
- Simplify the resulting rational expression.
Example: Subtract and simplify .
Solution.
Adding or Subtracting Rational Expressions with Different Denominators
To add or subtract rational expressions with different denominators, we need to rewrite the rational expressions to equivalent rational expressions with the same denominator, say the LCD.
Equivalent Reduction.
What if all denominators are the same? How to make denominators the same? Reducing the problem to an easier one using equivalent operations helps solve the problem.
Example: Find the LCD of and
.
Solution.
Factor each denominator.
List the factors of the first denominator and add unlisted factors of the second factor to get the final list.
First list Second list Final list The LCD is the product of factors in the final list.
Example: Subtract and simplify
Solution.
- Find the LCD.
First factor denominators.
Then using the table to find the final list of factors of the LCD.
First list Second list Final list
.
- Rewrite each rational expression into an equivalent expression with the LCD as the new denominator.
- Subtract and simplify.
Simplifying Complex Rational Expressions
A complex rational expression is a rational expression whose denominator or numerator contains a rational expression.
A complex rational expression is equivalent to the quotient of its numerator by its denominator. That suggests the following strategy to simplify a complex rational expression.
Simplify and Change the Viewpoint. A complex rational expression is a quotient of two rational expressions. You may rewrite it as an multiplication by flipping the denominator. However, it’s better to simply the numerator and denominator or you won’t see a good looking new expression.
Example: Simplify
Solution.
- Simplify the numerator and the denominator.
- Rewrite as a product.
- Multiply and simplify.
Another way to simplify a complex rational expression is to multiply the LCD to both the denominator and numerator and then simplify.
Example: Simplify
Solution.
- Find the LCD of all denominators.
In this case, the LCD is.
- Multiply the complex rational expression by
and simplify.
Practice
Exercise: Simplify.
Exercise: Multiply and simplify.
Exercise: Divide and simplify.
Exercise: Simplify.
Exercise: Add/subtract and simplify.
Exercise: Find the LCD of rational expressions.
and
and
Exercise: Add and simplify.
Exercise: Subtract and simplify.
Exercise: Simplify.
Exercise: Subtract and simplify.
Exercise: Simplify.
Exercise: Simplify.
Exercise: Simplify.
Exercise: Tim and Jim refill their cars at the same gas station twice last month. Each time Tim got $20 gas and Jim got 8 gallon. Suppose they refill their cars on same days. The price was $2.5 per gallon the first time. The price on the second time changed. Can you find out who had the better average price?
Radicals and Rational Exponents
Do You Want to Be a Fire Fighter
To reach the 5th floor window of a building that is 25 feet from the location of the turntable aerial ladder truck. How long should the ladder be placed to reach the window? The hight of that window is 50 feet.
Radical Expressions
If , then we say that
is a square root of
. We denote the positive square root of
as
, called the principal square root.
For any real number , the expression
can be simplified as
If , then we say that
is a cube root of
. The cube root of a real number
is denoted by
.
For any real number , the expression
can be simplified as
In general, if , then we say that
is an
-th root of
. If
is even, the positive
-th root of
, called the principal
-th root, is denoted by
. If
is odd, the
-the root
of
has the same sign with
.
In , the symbol
is called the radical sign,
is called the radicand, and
is called the index.
If is even, then the
-th root of a negative number is not a real number.
For any real number , the expression
can be simplified as
if
is even.
if
is odd.
A radical is simplified if the radicand has no perfect power factors against the radical.
Example: Simplify the radical expression using the definition.
Solution.
.
.
Rational Exponents
If is a real number, then we define
as
Rational exponents have the same properties as integral exponents:
Example: Simplify the radical expression or the expression with rational exponents. Write in radical notation.
Solution.
In general, rewriting radical in rational exponents helps simplify calculations.
Product and Quotient Rules for Radicals
If and
are real numbers, then
If and
are real numbers and
, then
Example: Simplify the expression.
.
.
Solution.
.
.
Combining Like Radicals
Two radicals are called like radicals if they have the same index and the same radicand. We add or subtract like radicals by combining their coefficients.
Example: Simplify the expression.
Solution.
Multiplying Radicals
Multiplying radical expressions with many terms is similar to that multiplying polynomials with many terms.
Example: Simplify the expression.
Solution.
Rationalizing Denominators
Rationalizing denominator means rewriting a radical expression into an equivalent expression in which the denominator no longer contains radicals.
Example: Rationalize the denominator.
Solution.
- In this case, to get rid of the radical in the bottom, we multiply the expression by
so that the radicand in the bottom becomes a perfect power.
- In this case, we use the formula
. Multiply the expression by
.
Complex Numbers
The imaginary unit is defined as
. Hence
.
If is a positive number, then
.
Let and
are two real numbers. We define a complex number by the expression
. The number $a $ is called the real part and the number
is called the imaginary part. If
, then the complex number
is just the real number. If
, then we call the complex number
an imaginary number. If
and
, then the complex number
is called a purely imaginary number.
Adding, subtracting, multiplying, dividing or simplifying complex numbers are similar to those for radical expressions. In particular, adding and subtracting become similar to combining like terms.
Example: Simplify and write your answer in the form , where
and
are real numbers and
is the imaginary unit.
Solution.
Example: Evaluate the express for
. Write your answer in the form
.
Solution.
Practice
Exercise: Evaluate the square root. If the square root is not a real number, state so.
Exercise: Simplify the radical expression.
Exercise: Simplify the radical expression.
Exercise: Simplify the radical expression. Assume all variables are positive.
Exercise: Write the radical expression with rational exponents.
Exercise: Write in radical notation and simplify.
Exercise: Simplify the expression. Write with radical notations. Assume all variables represent nonnegative numbers.
Exercise: Simplify the expression. Write in radical notation. Assume is nonnegative.
Exercise: Simplify the expression. Write in radical notation. Assume is nonnegative.
Exercise: Simplify the expression. Write in radical notation. Assume all variables are nonnegative.
Exercise: Multiply and simplify.
Exercise: Simplify the radical expression. Assume all variables are positive.
Exercise: Divide. Assume all variables are positive. Answers must be simplified.
Exercise: Add or subtract, and simplify. Assume all variables are positive.
Exercise: Add or subtract, and simplify. Assume all variables are positive
Exercise: Multiply and simplify. Assume all variables are positive.
Exercise: Multiply and simplify. Assume all variables are positive.
Exercise: Simplify the radical expression and rationalize the denominator. Assume all variables are positive.
Exercise: Simplify the radical expression and rationalize the denominator. Assume all variables are positive.
Exercise: Simplify and rationalize the denominator. Assume all variables are positive.
Exercise: Add, subtract, multiply complex numbers and write your answer in the form .
Exercise: Add, subtract, multiply complex numbers and write your answer in the form .
Exercise: Divide the complex number and write your answer in the form .
Exercise: Simplify the expression.
Exercise: Evaluate the polynomial for
. Write your answer in the form
.
Part 2: Equations and Applications
Solving Polynomial Equations by Factoring
Handshaking Problem
In meeting room, a group of people all shook hands with one another. In total, 15 handshakes occurred. Do you know how many people in the group?
Properties of Equations
An equation is an statement that asserts an equality containing unknown variables. For example, is an equation of the unknown variable
.
Equations often contain variables other than the unknowns. Those variables, which are assumed to be known, are usually called constants, coefficients or parameters. For example, in the linear equation of (the unknown)
, the variables
,
and
are referred as known coefficients or constants.
An identity is an equation that is true for all possible values of the variable(s) it contains. For example, is an identity.
Solving an equation consists of determining values of the variables that make the equality true. Two equations are said to be equivalent if and only if they have the same solution set, that is, a solution of one equation is also a solution of the other equation. For example and
are equivalent.
When solving an equation, the following operations can be used to transform an equation to an equivalent one:
- Adding or subtracting the same quantity to both sides of an equation. For example,
is equivalent to
.
- Multiplying or dividing both sides of an equation by a non-zero quantity. For example
is equivalent to
.
- Applying an identity to transform one side of the equation. For example,
is equivalent to
, where the identity
was applied.
In general, one may apply any choice of a function to both sides of the equation to make a transformation. The resulting equation still has the solutions of the original equations as it solutions. However, the resulting equation may also have some extra solutions which are called extraneous solutions. For example, taking squares of both sides of the equation produces the equation
. The new equation
has two solutions
and
, but the original equation
only has one solution. The solution
of the equation
is an extraneous solution of the equation
.
Quadratic Equations
A polynomial equation is an equation that can be written in the form where
is a positive integer and
.
A polynomial equation is called a quadratic equation if . For example,
. We often write a quadratic equation in its standard form
where
,
and
are numbers, and
.
When solving linear equations, arithmetic operations are enough. In general, one may need to use identity or functional operation. Factoring is one of those frequently used identity operation. Indeed, to solve a problem, a general strategy is to to reduce the original problem to easier problems. Using factoring and the zero product property: one can transform a polynomial equation into smaller degree polynomial equations. In particular, if
, then a solution of the quadratic equation
is a solution of either
or
.
Example: Solve the equation
Solution.
Rewrite the equation into “Expression=0” form and factor.
Apply the zero product property.
Solve each equation.
The solution set is
.
Example: Solve the equation
Solution.
- Rewrite the equation into “Expression=0” form and factor.
- Apply the zero product property.
- Solve each equation.
- The solution set is
.
Example: A rectangular garden is surrounded by a path of uniform width. If the dimension of the garden is meters by
meters and the total area is 216 square meters, determine the width of the path.
Solution.
- Suppose that the width of the frame is
meters. Translate given information into expressions in
and build an equation.
Total Width:Total Length:
WidthLength=Total Area:
- Solve the equation.
- So the width of the path is
meter.
Understand the Problem
When solving a word problem, you may first outline what’s known and what’s unknown, and restate the problem using algebraic expressions. Once you reformulated the problem algebraically, you may solve it using your mathematical knowledge.
Practice
Exercise: Solve the equation by factoring.
Exercise: Find all real solutions of the equation by factoring.
Exercise: A paint measuring inches by
inches is surrounded by a frame of uniform width. If the combined area of the paint and the frame is
square inches, determine the width of the frame.
Exercise: A rectangle whose length is meters longer than its width has an area
square meters. Find the width and the length of the rectangle.
Exercise: The product of two consecutive negative odd numbers is . Find the numbers.
Exercise: In a right triangle, the long leg is 2 inches more than double of the short leg. The hypotenuse of the triangle is 1 inch longer than the long leg. Find the length of the shortest side.
Exercise: A ball is thrown upwards from a rooftop. It will reach a maximum vertical height and then fall back to the ground. The height of the ball from the ground after time
seconds is
feet. How long it will take the ball to hit the ground?
Exercise: A toy factory estimates that the demand of a particular toy is units each week if the price is $
dollars per unit. Each week there is a fixed cost $40,000 to produce the demanded toys. The weekly revenue is a function of the price given by
- Find the function that models the weekly revenue,
, received when the selling price is $
per unit.
- What the price range so the the revenue is nonnegative.
Quadratic Formula
Estimate a Square Root
Can you estimate the irrational numbers ,
,
and
without using a calculator?
Can you estimate the square root , where
and
are positive integers?
Completing the Square
The square root property:
Suppose that . Then
or
, or simply
.
The square root property provides another method to solve a quadratic equation, completing the square. This method is based on the following observations: and more generally, let
, and
, then
The procedure to rewrite a trinomial as the sum of a perfect square and a constant is called completing the square.
Example: Solve the equation .
Solution.
- Isolate the constant.
- With
, add
to both sides to complete a square for the binomial
.
- Solve the resulting equation using the square root property.
Note that the solution can also be written as .
Example: Solve the equation .
Solution.
- Isolate the constant.
- Divide by
to rewrite the equation in
form
- With
, add
to both sides to complete the square for the binomial
.
- Solve the resulting equation and simplify.
Another way to complete the square is to use the formula , where
is the value of the polynomial
at
.
The Quadratic Formula
Using the method of completing the square, we obtain the follow quadratic formula for the quadratic equation with
:
The quantity is called the discriminant of the quadratic equation.
- If
, the equation has two real solutions.
- If
, the equation has one real solution.
- If
, the equation has two imaginary solutions (no real solutions).
Example: Determine the type and the number of solutions of the equation .
Solution.
- Rewrite the equation in the form
.
- Find the values of
,
and
.
- Find the discriminant
.
The equation has two imaginary solutions.
Example: Solve the equation .
Solution.
- Find the values of
,
and
.
- Find the discriminant
.
- Apply the quadratic formula and simplify.
Example: Find the base and the height of a triangle whose base is three inches more than twice its height and whose area is square inches. Round your answer to the nearest tenth of an inch.
Solution.
- We may suppose the height is
inches. The base can be expressed as
inches.
- By the area formula for a triangle, we have an equation.
- Rewrite the equation in
form.
- By the quadratic formula, we have
Since can not be negative,
and
. The height and base of the triangle are approximately
inches and 6.2 inches respectively.
Practice
Exercise: Solve the quadratic equation by the square root property.
Exercise: Solve the quadratic equation by completing the square.
Exercise: Determine the number and the type of solutions of the given equation.
Exercise: Solve using the quadratic formula.
Exercise: Solve using the quadratic formula.
Exercise: A triangle whose area is square meters has a base that is one meter less than triple the height. Find the length of its base and height. Round to the nearest hundredth of a meter.
Exercise: A rectangular garden whose length is feet longer than its width has an area 66 square feet. Find the dimensions of the garden, rounded to the nearest hundredth of a foot.
Rational Equations
A Problem of the Father of Algebra
The father of algebra, Muḥammad ibn Musa al-Khwarizmi, in his book “Algebra”, answered the following problem:
“… I have divided ten into two parts; and have divided the first by the second, and the second by the first, and the sum of the quotient is two and one-sixth;…”
Do you know what are those two parts?
Solving Rational Equations
A rational equation is an equation that contains a rational expression. One way to solve rational equations is to clear all denominators by multiplying the LCD to both sides. Note that because there are values such that the LCD has the value zero. Clearing denominator in general is not an equivalent transformation, rather a functional transformation. So don’t forget to check possible extraneous solutions.
Example: Solve
Solution.
- Find the LCD. Since
, the LCD is
.
- Clear denominators. Multiply each rational expression in both sides by
and simplify:
- Solve the resulting equation.
- Check for any extraneous solution by plugging the solution into the LCD to see if it is zero. If it is zero, then the solution is extraneous.
So
is a valid solution of the original equation.
Reduction With Auxiliary Conditions
Clearing denominator uses the strategy “reduction with auxiliary conditions”. The auxiliary condition used when clearing the denominators is that the LCD is non-zero. Generally, if you don’t know how to solve a problem under the given condition, you may try solve the problem by adding extra conditions first and then try to eliminate the extra conditions and/or their consequences.
Literal Equations
A literal equation is an equation involving two or more variables. When solving a literal equation for one variable, other variables can be viewed as constants.
Example: Solve for from the equation
Solution.
- The LCD is
.
- Clear denominators.
- Solve the resulting equation.
- The solution is
if
. If
, the equation has no solution.
Another way to solve a rational equation is to rewrite and simplify the equation into the form where
is a reduced fraction. Then the rational equation is equivalent to the equation
.
Practice
Exercise: Solve.
Exercise: Solve.
Exercise: Solve a variable from a formula.
- Solve for
from
.
- Solve for
from
.
Exercise: Solve for from the equation.
.
.
Exercise: David can row 3 miles per hour in still water. It takes him 90 minutes to row 2 miles upstream and then back. How fast is the current?
Exercise: The size of a A0 paper is defined to have an area of 1 square meter with the longer dimension meters. Successive paper sizes in the series A1, A2, A3, and so forth, are defined by halving the preceding paper size across the larger dimension. Can you find the dimension of a A4 paper?
Radical Equations
Design a Pendulum clock
A pendulum clock is a clock that uses a pendulum, a swinging weight, as its timekeeping element. Galileo Galilei discovered in early 17th century the relation between the length of a pendulum and the period
of th pendulum. For a pendulum clock, the relations is approximately determined by the following rule of thumb formula:
given that
and
are measured in meters and seconds respectively. If the period of a pendulum clock is 2 seconds, how long should be the pendulum?
Solving Radical Equations by Taking a Power
The idea to solve a radical equation is to first take
-th power of both sides to get rid of the radical sign, that is
and then solve the resulting equation.
Solve by Reduction
The goal to solve a single variable equation is to isolate the variable. When an equation involves radical expressions, you can not isolate the variable arithmetically without eliminating the radical sign unless the radicand is a perfect power. To remove a radical sign, you make take a power. However, you’d better to isolate it first. Because simply taking powers of both sides may create new radical expressions.
Example: Solve the equation
Solution.
Arrange terms so that one radical is isolated on one side of the equation.
Square both sides to eliminate the square root.
Solve the resulting equation.
Check all proposed solutions.
Pluginto the original equation, we see that the left hand side is
which is not equal to the right hand side. So
cannot be a solution.
Plug
into the original equation, we see that the left hand side is
. So
is a solution.
Example: Solve the equation
Solution.
- Isolated one radical.
- Square both sides to remove radical sign and then isolate the remaining radical.
- Square both sides to remove the radical sign and then solve.
Since
and
,
is a valid solution. Indeed,
Example: Solve the equation
Solution.
- Isolated the radical.
- Cube both sides to eliminate the cube root and then solve the resulting equation.
The solution is
.
Equations Involving Rational Exponents
Equation involving rational exponents may be solved similarly. However, one should be careful with meaning of the expression . When
is even and
is odd,
. Otherwise,
.
Example: Solve the equation .
Solution.
Since there are more than one term involving rational exponents, to solve the equation, we isolate one term and taking power and so on so forth. Check:
So the equation has one solution .
Example: Solve the equation .
Solution.
There are different way to solve this equation. One may is to take rational powers of both sides and solve the resulting equation. Check:
So the equation has two solutions and
.
Learn from Mistakes
Example: Can you find the mistakes made in the solution and fix it?
Solve the radical equation.
Solution (incorrect): Answer: the equation has two solutions
and
.
Solution.
When squaring one side of the equation, the other side as a whole should be squared. The mistake occurred at the squaring step. The right way to solve the equation is as follows. Because squaring is not an equivalent transformation in general, the solutions of the resulting equations must be checked. When
, the left side of the original equation is
. When
, the left side is
. So both
and
are solutions of the function
.
Practice
Exercise: Solve each radical equation.
Exercise: Solve each radical equation.
Exercise: Solve each radical equation.
Exercise: Solve each radical equation.
Exercise: Solve each radical equation.
Exercise: Solve each radical equation.
Absolute Value Equations
The Direction of a Number
Can you determine the value of the expression for all nonzero real number
and explain the meaning of the value?
Properties of Absolute Values
The absolute value of a real number , denoted by
, is the distance from
to
on the number line. In particular,
is always greater than or equal to
, that is
. Absolute values satisfy the following properties:
An absolute value equation may be rewritten as , where
represents an algebraic expression.
If is positive, then the equation
is equivalent to {
or
.}
If is negative, then the solution set of
is empty. An empty set is denoted by
.
More generally, is equivalent to
or
.
The equation is equivalent to
.
Example: Solve the equation
Solution.
The equation is equivalent to
The solutions are or
. In set-builder notation, the solution set is
.
Example: Solve the equation
Solution.
Rewrite the equation into
form.
Solve the equation.
The solutions are or
. In set-builder notation, the solution set is
.
Example: Solve the equation
Solution.
Rewrite the equation into
form.
Solve the equation.
The solutions are or
. In set-builder notation, the solution set is
.
Example: Solve the equation
Solution.
Rewrite the equation into
form.
Solve the equation.
The solutions are or
. In set-builder notation, the solution set is
.
Example: Solve the equation
Solution.
Note that two numbers have the same absolute value only if they are the same or opposite to each other. Then the equation is equivalent to
The solutions are and
. In set-builder notation, the solution set is
.
Example: Solve the equation
Solution.
Since is positive,
. Moreover, because
, the equation is equivalent to
The original equation only has one solution . In set-builder notation, the solution set is
.
Practice
Exercise: Find the solution set for the equation.
Exercise: Find the solution set for the equation.
Exercise: Find the solution set for the equation.
Exercise: Find the solution set for the equation.
Exercise: Find the solution set for the equation.
Exercise: Find the solution set for the equation.
Linear Inequalities
Know the Grade You Must Earn
- A course has three types of assessments: homework, monthly test and the final exam. The grading policy of the course says that homework counts 20%, monthly test counts 45% and the final exam counts for 35%. At the last day of class a student wants to know the minimum grade needed on the final to get a grade C or better, equivalently, overall grade 74 or above. The student earned 100 on homework and 80 on monthly test.
- What the minimum grade the student must earn on the final to get a C or better?
- If, in addition, the final exam must be at least 55 to earn a C or better, what would be the minimum grade needed?
- The college student has attempted 30 credits and a cumulated GPA 1.8. To graduate from the college, the GPA must be 2.0 or higher and the total credits must be at least 60. Now the student decides to spend more time on studying and aims at an cumulated GPA 2.5 on further courses.
How many more attempted credits the student must earn to graduate?
Cumulated GPA =
Total Quality Points Earned = Sum of
Properties and Definitions
Properties of Inequalities
An inequality defines a relationship between two expressions. The following properties show when the inequality relationship is preserved or reversed.
Property | Example |
---|---|
The additive property If If | If Simplifying both sides, we get |
The positive multiplication property If If | If Simplifying both sides, we get |
The negative multiplication property If If | If If Simplifying both sides, we get |
These properties also apply to ,
and
.
It’s always better to view as
. Because addition has the commutative property.
Compound Inequalities
A compound inequality is formed by two inequalities with the word and or the word or. For examples, the following are three commonly seen type compound inequalities:
The third compound inequality is simplified expression for the compound inequality
and
.
Interval Notations
Solutions to an inequality normally form an interval which has boundaries and should reflect inequality signs. Depending on the form of an inequality, we may a single interval and a union of intervals. For example, suppose , we have the following equivalent representations of inequalities.
Examples
Think backward.
To solve a problem, knowing what to expect helps you narrow down the gap step by step by comparing the goal and your achievement.
An inequality (equation) is solved if the unknown variable is isolated. That’s what to be expected. To isolate the unknown variable, you use comparisons to determine what mathematical operations should be applied. When an operation is applied to one side, the same operation should also be applied to the other side. For inequalities, we also need to determine whether the inequality sign should be preserved or reversed according to the operation.
Example: Solve the linear inequality
Solution.
Example: Solve the linear inequality
Solution.
The solution set is
.
Example: Solve the compound linear inequality
Solution.
That is
. The solution set is
.
Example: Solve the compound linear inequality
Solution.
That is
or
. The solution set is
.
Example: Solve the compound linear inequality
Solution.
The solution set is
.
Example: Solve the compound linear inequality
Solution.
The solution set is
.
Example: Suppose that . Find the range of
. Write your answer in interval notation.
Solution.
To get from
, we need first multiply
be
and then add
.
The range of
is
.
Understand the Problem.
Understanding the known, the unknown and the condition of the given problem is the key to solve the problem. Normally, by comparing the known and unknown, you will find the way to solve the problem.
Practice
Exercise: Solve the linear inequality. Write your answer in interval notation.
Exercise: Solve the linear inequality. Write your answer in interval notation.
Exercise: Solve the compound linear inequality. Write your answer in interval notation.
Exercise: Solve the compound linear inequality. Write your answer in interval notation.
Exercise: Solve the compound linear inequality. Write your answer in interval notation.
Exercise: Solve the compound linear inequality. Write your answer in interval notation.
Exercise: Solve the linear inequality. Write your answer in interval notation.
Exercise: Solve the compound linear inequality. Write your answer in interval notation.
Exercise: Suppose . Find the range of
. Write your answer in interval notation.
Exercise: Suppose that and
. Find the range of
. Write your answer in interval notation.
Exercise: A toy store has a promotion “Buy one get the second one half price” on a certain popular toy. The sale price of the toy is $20 each. Suppose the store makes more profit when you buy two. What do you think the store’s purchasing price of the toy is?
Part 3: Functions and Applications
Introduction to Functions
It is a Flu Season
The following graph shows a relation between the week number from 2019/09/30 to 2020/01/27, and the number of people being tested for flu, and the number of people whose tests were positive.
- Can you describe this relation?
- Can you draw conclusions based on the graph?
- Can you estimate how many people in total had positive tests on 2020/01/01?
- What do think the trending will be after 2020/1/27? Why?
Definition and Notations
A relation is a set of ordered pairs. The set of all first components of the ordered pairs is called the domain. The set of all second components of the ordered pairs is called the range.
A function is a relation such that each element in the domain corresponds to exactly one element in the range.
For a function, we usually use the variable to represent an element from the domain and call it the independent variable. The variable
is used to represent the value corresponding to
and is called the dependent variable. We say
is a function of
. When we consider several functions together, to distinguish them we named functions by a letter such as
,
, or
. The notation
, read as “
of
” or “
at
”, represents the output of the function
when the input is
.
The domain of a function is the set of all allowed inputs. The range of a function is the set of all outputs.
To find the value of a function define by an equation at a given number, we substitute the independent variable by the given number and then evaluate the expression. We call the procedure evaluating a function.
Example: Find the indicated function value.
,
,
,
.
Solution.
.
.
.
Graphs of Functions
The graph of a function is the graph of its ordered pairs. A graph of ordered pairs in the rectangular coordinate system defines
as a function of
if any vertical line crosses the graph at most once. This test is called the vertical line test.
Example: Determine which of the following graphs defines a function.
Solution.
Because in graphs A. B. C. there are vertical lines intersecting the graph at two points. So those graphs fail the vertical line test and hence don’t define functions. In graph D., although the graphs are not connected, but any vertical line only intersects one point. Therefore, Graph D. defines a functions.
Graph Reading
The domain of a graph is the set of -coordinates of all points on the graph. The range of a graph is the set of
-coordinates of all points on the graph. To find the domain of a graph, we look for the left and the right endpoints. To find the range of a graph, we look for the highest and the lowest positioned points.
To find the coordinates of a point on a graph, one draw a horizontal line and a vertical line through the point. The number on the -axis where the vertical line passing through is the
-coordinate of the point. The number on the
-axis where the horizontal line passing through is the
-coordinate of the point.
Example: Use the graph in the picture to answer the following questions.
- Determine whether the graph is a function and explain your answer.
- Find the domain (in interval notation) of the graph.
- Find the range (in interval notation) of the graph.
- Find the interval where the graph is above
.
- Find the interval where the graph is is decreasing.
- Find all maximum and minimum values of the function if they exist.
- Find the value of
such that the point
is on the graph.
- Find the value of
such that
is on the graph.
Solution.
- The graph is a function. Because every vertical line crosses the graph at most once.
- The graph has the left endpoint at
and but no right endpoint. So the domain is
.
- The graph has a lowest positioned point
but no highest positioned point. So the range is
.
- The graph is above 2 over the interval
.
- The graph is decreasing over the interval
.
- The graph has minima at
and
.
- The
-value of the point
on the graph is
.
- The
-value of the point
on the graph is
.
Practice
Exercise: Find the indicated function values for the functions and
. Simplify your answer.
Exercise: Suppose .
- Compute
- Compute
Exercise: Suppose the domain of the linear function is
. Find the range of the function.
Exercise: Use the graph in the picture to answer the following questions.
- Determine whether the graph is a function and explain your answer.
- Find the domain of the graph (write the domain in interval notation).
- Find the range of the graph (write the range in interval notation).
- Find the interval where the graph is above the
-axis.
- Find all points where the graph reaches a maximum or a minimum.
- Find the values of the
-coordinate of all points on the graph whose
-coordinate is
.
Exercise: Use the graph of the function in the picture to answer the following questions.
- Find the
-intercept.
- Find the value
- Find the values
such that
.
- Find the solution to the inequality
. Write in interval notation.
Exercise: Today Matt drove from home to school in 30 minutes. He spent 6 minutes on local streets before driving on the highway and 4 minutes on local streets towards school after getting off the highway. On local streets, his average speed is 30 miles per hour. On the highway, his average speed is 60 miles per hours.
- Write the distance
(in miles) he drove as a function of the time
(in minutes)?
- After 15 minutes, where was he and how far did he drive?
- How far did he drive from home to school?
Linear Functions
Cost, Revenue and Profit
A company has fixed costs of $10,000 for equipment and variable costs of $15 for each unit of output. The sale price for each unit is $25. What is total cost, total revenue and total profit at varying levels of output?
The Slope-Intercept Form Equation
The slope of a line measures the steepness, in other words, “rise” over “run”, or rate of change of the line. Using the rectangular coordinate system, the slope of a line is defined as
where
and
are any two distinct points on the line. If the line intersects the
-axis at the point
, then a point
is on the line if and only if
This equation is called the slope-intercept form of the line.
Point-Slope Form Equation of a Line
Suppose a line passing through the point has the slope
. Solving from the slope formula, we see that any point
on the line satisfies the equation equation
which is called the point-slope form equation.
Linear Function
A linear function is a function whose graph is a line. An equation for
can be written as
where
is the slope and
.
A function is a linear function if the following equalities hold
for any three distinct points
,
and
on the graph of
.
Equations of Linear Functions
Example: Find the slope-intercept form equation for the linear function such that
and
.
Solution.
- Find the slope
:
.
- Plug one of the points, say
in the point-slope form equation, we get
- Simplify the above equation, we get the slope-intercept form equation
.
Graph a Linear Function by Plotting Points
Example: Sketch the graph of the linear function .
Solution.
Method 1: Get points by evaluating .
- Choose two or more input values, e.g.
and
.
- Evaluate
:
and
.
- Plot the points
and
and draw a line through them.
Method 2: Get points by raise and run.
- Plot the
-intercept
.
- Use
to get one or more points, e.g, we will get
by taking
and
, i.e. move up
unit, then move to the right
units.
- Plot the points
and
and draw a line through them.
Horizontal and Vertical Lines
A horizontal line is defined by an equation . The slope of a horizontal line is simply zero. A vertical line is defined by an equation
. The slope of a vertical line is undefined.
A vertical line gives an example that a graph is not a function of . Indeed, the vertical line test fails for a vertical line.
Explicit Function
When studying functions, we prefer a clearly expressed function rule. For example, in , the expression
clearly tells us how to produce outputs. For a function
defined by an equation, for instance,
, to find the function rule (that is an expression), we simply solve the given equation for
.
Now, we get
.
Perpendicular and Parallel Lines
Any two vertical lines are parallel. Two non-vertical lines are parallel if and only if they have the same slope.
A line that is parallel to the line has an equation
, where
.
Horizontal lines are perpendicular to vertical lines. Two non-vertical lines are perpendicular if and only if the product of their slopes is .
A line that is perpendicular to the line has an equation
.
Finding Equations for Perpendicular or Parallel Lines
Example: Find an equation of the line that is parallel to the line and passes through the point
.
Solution.
- Find the slope
of the original line from the slope-intercept form equation by solving for
.
. So
.
- Find the slope
of the parallel line.
- Use the point-slope form.
Example: Find an equation of the line that is perpendicular to the line and passes through the point
.
Solution.
- Find the slope
of the original line from the slope-intercept form equation by solving for
.
. So
.
- Find the slope
of the perpendicular line.
- Use the point-slope form.
Practice
Exercise: Find the slope of the line passing through
and
and
.
Exercise: Find the point-slope form equation of the line with slope that passes though
.
Exercise: Find the point-slope form equation of the line passing thought and
.
Exercise: Find the slope-intercept form equation of the line passing through and
.
Exercise: Determine whether the linear functions and
with the following values
and
define the same function. Explain your answer.
Exercise: Suppose the points and
are on the graph of a linear function
. Find
.
Exercise: Graph the functions.
Exercise: A storage rental company charges a base fee of $15 and $ per day for a small cube. Suppose the cost is $20 dollars for 10 days.
- Write the cost
(in dollars) as a linear function of the number of days
.
- How much would it cost to rent a small cube for a whole summer (June, July and August)?
Exercise: Find an equation for each of the following two lines which pass through the same point .\
- The vertical line.
- The horizontal line.
Exercise: Line is defined by the equation
. What is the slope
of the line that is parallel to the line
? What is the slope
of the line that is perpendicular to the line
.
Exercise: Line is defined by
. Line
passes through
and
. Determine whether
and
are parallel, perpendicular or neither.
Exercise: Find the point-slope form and then the slope-intercept form equations of the line parallel to and passing through the point
.
Exercise: Find the slope-intercept form equation of the line that is perpendicular to and passing through the point
Exercise: The line is defined
. The line
is defined by the equation
. The line
is defined by
. Determine whether
,
and
are parallel or perpendicular to each other.
Exercise: Use the graph of the line to answer the following questions
- Find an equation for the line
.
- Find an equation for the line
perpendicular to
and passing through
.
- Find an equation for the line
parallel to
and passing through
.
Exercise: Determine whether the points ,
,
and
form a square. Please explain your conclusion.
Exercise: A tutoring center has fixed monthly costs of $5000 that covers rent, utilities, insurance, and advertising. The center charges $60 for each private lesson and each lesson has a variable cost of $35 to pay the instructor. Determine the number of private lessons that must be held for the tutoring center to make a profit.
Exercise: In 2008, an elementary school population was 1011. By 2018 the population had grown to 1281. Assume the population changes linearly.
Find an equation for the population
of the school
years after 2008.
What would be the population of the school in 2021.
Quadratic Functions
Maximize the Revenue
When price increases, demand decreases and vice verse. A retail store found that the price as a function of the demand
for a certain product is
. The revenue
of selling
units is
. To maximize the revenue, what should be the price?
The Graph of a Quadratic Function
The graph of a quadratic function ,
, is called a parabola.
By completing the square, a quadratic function can always be written in the form
, where
and
.
- The line
is called the axis of symmetry of the parabola.
- The point
is called the vertex of the parabola.
The Minimum or Maximum of a Quadratic Function
Consider the quadratic function ,
.
- If
, then the parabola opens upward and
has a minimum
at the vertex.
- If
, then the parabola opens downward and
has a maximum
at the vertex.
Intercepts of a Quadratic Function
Consider the quadratic function ,
.
- The
-intercept is
.
- The
-intercepts, if exist, are the solutions of the equation
.
Example: Does the function have a maximum or minimum? Find it.
Solution.
- Since
, the function opens upward and has a minimum.
- Find the line of symmetry
:
.
- Find the minimum by plugging
into the function
. The minimum is
Example: Consider the function . Find values of
such that
.
Solution.
- Set up the equation for
. [-x^2+3x+6=2]
- Solve the equation
, we get
or
. The values of
such that
are
and
.
Example: A quadratic function whose the vertex is
has a
-intercept
. Find the equation that defines the function.
Solution.
- Write down the general form of
using only the vertex. Quadratic functions with the vertex at
are defined by
, where
is a nonzero real number.
- Determine the unknown
using the remaining information. Since
is on the graph of the function, the number
must satisfy the equation
.
- Solving for
from the equation, we get
. The quadratic function
is given by
.
Practice
Exercise: Sketch the graph of the quadratic functions and find
- the coordinates of the
-intercepts,
- the coordinates of the
-intercept,
- the equation of the axis of symmetry,
- the coordinates of the vertex.
- the interval of
values such that
.
Exercise: Sketch the graph of the quadratic functions and find
- the coordinates of the
-intercepts,
- the coordinates of the
-intercept,
- the equation of the axis of symmetry,
- the coordinates of the vertex.
- the interval of
values such that
.
Exercise: Consider the parabola in the graph.
- Determine the coordinates of the
-intercepts.
- Determine the coordinates of the
-intercept.
- Determine the coordinates of the vertex.
- For what values of
is
.
- Find an equation for the function.
Exercise: Consider the graph of the function shown in the picture.
- Determine the coordinates of the
-intercepts.
- Determine the coordinates of the
-intercept.
- Determine the coordinates of the vertex.
- Find the domain of the function.
- Find the range of the function.
- For what values of
is
.
- Over which interval is the function
positive.
- Over which interval is the function
decreasing.
- Find an additional point on the graph.
- Find an equation for the function.
Exercise: Consider the quadratic functions and find the following values or intervals
- the coordinates of all intercepts
- the coordinates of the vertex
- the equation and graph of the axis of symmetry
- the domain and range in interval notation
- the coordinates of an additional point on the graph
- the maximum or minimum value
- the value at which the max or min is reached
- the interval over which the function is negative
- the interval over which the function is positive
- the interval over which the function is increasing.
Exercise: A store owner estimates that by charging dollars each for a certain cell phone case, he can sell
phone cases each week. The revenue in dollars is
when the selling price of a computer is
, Find the selling price that will maximize revenue, and then find the amount of the maximum revenue.
Exercise: A ball is thrown upwards from a rooftop. It will reach a maximum vertical height and then fall back to the ground. The height of the ball from the ground after time
seconds is
feet.
- When will the toy rocket reach its maximum height? What will be the maximum height?
- When will the toy rocket hit the ground?
- How high above the ground will the toy rocket be after 2 seconds.
- When will the toy rocket be 96 feet above the ground?
Exercise: A ball is thrown upward from the ground with an initial velocity ft/sec. The height
of the ball after
seconds is
. The ball hits the ground after 4 seconds. Find the maximum height and how long it will take the ball to reach the maximum height.
Exercise: A toy factory estimates that the demand of a particular toy is units each week if the price is $
dollars per unit. Each week there is a fixed cost $40,000 to produce the demanded toys. The weekly revenue is a function of the price given by
- Find the function that models the weekly revenue,
, received when the selling price is $
per unit.
- What the price range so the the revenue is nonnegative.
Rational Functions
The Law of Lever
“Give me a fulcrum and a place to stand, I will move the world.” by Archimedes of Syracuse
In volume I of his book “On the Equilibrium of Planes”, Archimedes proved that magnitudes are in equilibrium at distances reciprocally proportional to their weights. See the video Law of the Lever on youtube for an animated explanation.
Suppose there is a infinite long lever with a load 100 newton that is placed 1 meters away the fulcrum, the pivoting point.
- Can you find the force needed to balance the load in terms of the distance away from the fulcrum?
- How much force will be needed if it is place 5 meters away from the fulcrum?
The Domain of a Rational Function
A rational function is defined by an equation
, where
and
are polynomials and the degree of
is at least one. Since the denominator cannot be zero, the domain of
consists all real numbers except the numbers such that
Example: Find the domain of the function .
Solution.
Solve the equation , we get
. Then the domain is
. In interval notation, the domain is
Practice
Exercise: Find the domain of each function. Write in interval notation.
Radical Functions
Speed of a Tsunami
A tsunami is generally referred to is a series of waves on the ocean caused by earthquakes or other events that cause sudden displacements of large volumes of water. In ideal situation, the velocity of a wave at where the water depth is
meters is approximately
The wave will slow down when closer to the coast but will be higher.
Suppose a tsunami was caused by earthquake somewhere 10000 meters away the coast of California. The depth of the water where the tsunami was generated is 5000 meter.
- What’s the initial speed of the tsunami?
- What’s the speed of the tsunami at where the water depth is 2000.
- Suppose the speed wouldn’t decrease, how long it takes the tsunami reach the coast?
The Domain of a Radical Function
A radical function is defined by an equation
, where
is an algebraic expression. For example
. When
is odd number,
can be any real number. When
is even,
has to be nonnegative, that is
so that
is a real number.
Example: Find the domain of the function .
Solution.
Since the index is which is even, the function has real outputs only if the radicand
. Solve the inequality, we get
. In interval notation, the domain is
Practice
Exercise: Find the domain of each function. Write in interval notation.
Exponential Functions
Half-life
Half-life is the time required for a quantity to reduce to half of its initial value.
A certain pesticide is used against insects. The half-life of the pesticide is about 12 days. After a month how much would left if the initial amount of the pesticide is 10 g? Can you write a function for the remaining quantity of the pesticide after days?
More examples on exponential functions can be found on http://passyworldofmathematics.com/exponents-in-the-real-world/.
Definition and Graphs of Exponential Functions
Let be a positive number other than
(i.e.
and
). The exponential function
of
with the base
is defined as
Graphs of exponential functions:
The exponential function is a one-to-one function: any vertical line or any horizontal line crosses the graph at most once. Equivalently, the equation
has at most one solution for any real number
.
The Natural Number 
The natural number is the number to which the quantity
approaches as
takes on increasingly large values. Approximately,
.
Compound Interests
After years, the balance
in an account with a principal
and annual interest rate
is given by the following formulas:
- For
compounding periods per year:
.
- For compounding continuously:
.
Example: A sum of is invested at an annual rate of
, Find the balance, to the nearest hundredth dollar, in the account after
years if the interest is compounded
- monthly,
- quarterly,
- semiannually,
- continuous.
Solution.
- Find values of
,
,
and
. In this case,
,
,
and
depends compounding.
- Plug the values in the formula and calculate.
- ``Monthly’’ means
. Then
- ``Quarterly’’ means
. Then
- ``semiannually’’ means
. Then
- For continuously compounded interest, we have
In the compounded investment module, the is an approximation of the period interest rate. Indeed, if the period rate
satisfies the equation
, or equivalently
. Using the formula
, one may approximately replace
by
and obtain the approximation
.
Example: The population of a country was about 0.78 billion in the year 2015, with an annual growth rate of about 0.4%. The predicted population is billions after
years since 2015. To the nearest thousandth of a billion, what will the predicted population of the country be in 2030?
Solution.
The population is approximately
Practice
Exercise: The value of a car is depreciating according to the formula: , where
is the age of the car in years. Find the value of the car, to the nearest dollar, when it is five years old.
Exercise: A sum of $20,000 is invested at an annual rate of 5.5%, Find the balance, to the nearest dollar, in the account after 5 years subject to
- monthly compounding,
- continuously compounding.
Exercise: Sketch the graph of the function and find its range.
Exercise: Use the given function to compare the values of ,
and
and determine which value is the largest and which value is the smallest. Explain your answer.
Logarithmic Functions
Estimate the Number of Digits
Can you estimate the number of digits in the integer part of the number ?
Definition and Graphs of Logarithmic Function
For ,
and
, there is a unique number
satisfying the equation
. We denote the unique number
by
, read as logarithm to the base
of
. In other words, the defining relation between exponentiation and logarithm is
The function
is called the logarithmic function
of
with the base
.
Graphs of logarithmic functions:
Common Logarithms and Natural Logarithms
A logarithmic function with base 10 is called the common logarithmic function and denoted by
.
A logarithmic function with base the natural number
is called the natural logarithmic function and denoted by
.
Basic Properties of Logarithms
When and
, and
, we have
.
.
and
.
Example: Convert between exponential and logarithmic forms.
Solution.
When converting between exponential and logarithmic forms, we move the base from one side to the other side, then add or drop the log sign.
- Move the base 10 to the right side and drop the log from the left:
- Move the 3 to the right and add log the the right:
Example: Evaluate the logarithms.
Solution.
The key is to rewrite the log and the power so that they have the same base.
.
Example: Find the domain of the function .
Solution.
The function has a real output if . Solving the inequality, we get
. So the domain of the function is
.
Properties of Logarithms
For ,
,
and
, we have
- (The product rule)
- (The quotient rule)
.
- (The power rule)
, where
is any real number.
- (The change-of-base property)
, where
and
. In particular,
Example: Expand and simplify the logarithm .
Solution.
Example: Write the expression as a single logarithm.
Solution.
Example: Evaluate the logarithm and round it to the nearest tenth.
Solution.
On most scientific calculator, there are only the common logarithmic function LOG
and the natural logarithmic function LN
. To evaluate a logarithm based on a general number, we use the change-of-base property. In this case, the value of is
Example: Simplify the logarithmic expression
Solution.
Practice
Exercise: Write each equation into equivalent exponential form.
Exercise: Write each equation into equivalent logarithmic form.
Exercise: Evaluate.
Exercise: Evaluate.
Exercise: Find the domain of the function . Write in interval notation.
Exercise: Sketch the graph of each function and find its range.
Exercise: Expand the logarithm and simplify.
Exercise: Expand the logarithm and simplify.
Exercise: Write as a single logarithm.
Exercise: Write as a single logarithm.
.
Exercise: Evaluate the logarithm and round it to the nearest hundredth.
Exercise: Simplify the logarithmic expression
Applications of Exponential and Logarithmic Functions
Newton’s Law of Cooling
Suppose an object with an initial temperature is placed in an environment with surrounding temperature
. By Newton’s Law of Cooling, after
minutes, the temperature of the object
is given by the exponential function
where
is a positive constant characteristic of the system.
A cup of coffee is brewed with a temperature 195°F and placed in a room with the temperature 60°F. The cooling constant for a cup of coffee is $ r = .09 ^{-1}$.
- After 30 minutes, what is the temperature of the coffee?
- How long it takes for the coffee to cool down to the room temperature?
Exponential and Logarithmic Equations
To solve an exponential or logarithmic equation, the first step is to rewrite the equation with a single exponentiation or logarithm. Then we can use the equivalent relation between exponentiation and logarithm to rewrite the equation and solve the resulting equation.
Example: Solve the equation .
Solution.
- Rewrite the equation in the form
:
- Take logarithm of both sides and simplify:
- Solve the resulting equation:
Example: Solve the equation .
Solution.
- Rewrite the equation in the form
:
- Rewrite the equation in the exponential form (moving the base):
- Solve the resulting equation
. The solutions are
and
- Check proposed solutions. Both
and
has to be positive. So
is not a solution of the original equation. When
, we have
. So
is a solution.
Solving Compound Interest Model
Example: A check of $5000 was deposited in a savings account with an annual interest rate which is compounded monthly. How many years will it take for the money to raise by 20%?
Solution.
The question tells us the following information: ,
,
, and
. What we want to know is the number of years
. The compound interest model tells us that
satisfies the following equation:
This is an exponential equation and can be solve using logarithms.
So it takes about 3 years for the savings to raise by 20%.
When solving exponential and logarithmic equations, you may also use the one-to-one property if both sides are powers with the same base or logarithms with the same base.
Practice
Exercise: Solve the exponential equation.
Exercise: Solve the exponential equation.
Exercise: Solve the logarithmic equation.
Exercise: Solve the logarithmic equation.
Exercise: For the given function, find values of satisfying the given equation.
,
,
Exercise: Find intersections of the given pairs of curves.
and~
.
and~
Exercise: Using the formula to determine how many years, to the nearest hundredth, it will take to double an investment $20,000 at the interest rate 5% compounded monthly.
Exercise: Newton’s Law of Cooling states that the temperature of an object at any time
satisfying the equation
, where
is the the temperature of the surrounding environment,
is the initial temperature of the object, and
is positive constant characteristic of the system, which is in units of
. In a room with a temperature of
, a cup of tea of
was freshly brewed. Suppose that
. In how many minutes, the temperature of the tea will be
?
Exercise: A culture of bacteria began with 1000 bacteria and grows exponentially. An hour later there were 1320 bacteria. How many hours after starting will there be 2790 bacteria?
Exercise: A type of virus started spreading. After 10 days, 580 infected cases was confirmed. The natural log of the number of confirmed cases approximately is a linear function of the number of days after the virus started spreading. The slope of this linear function is 0.28. Suppose the pattern continues. Estimate the number of infected cases after 20 days.