Chapter 12: Inverse, Exp onential and Logarithmic
Functions
•Define an inverse function
•Look at the relationship between Exponential and Logarithmic functions
• Properties of Exponential and Logarithmic functions
• Common and natural log properties
•Applications of Exponential and Logarithmic functions
Section 12.1: Inverse Functions
•One-to-one functions
•Horizontal line test
• Equations of inverses
One-to-One Functions
•Recall that a function has a unique output for each input. In other words, for
each x value
there is a unique y value
•A one-to-one function is one in which only one x value corresponds to each y
value and
only one y value corresponds to each x value
Inverses of Functions
•When using a table of values or coordinate points, the inverse of a one-to-one
function is
found by switching the domain and range
•An inverse of a function is found by interchanging the x values and the y
values
•The inverse of a one-to-one function, f, is written as f -1
•Since the inverse is formed by interchanging the domain and range, the domain
of f
becomes the range of f -1and the range of f becomes the domain of
f -1
Examples: Decide whether each function is
one-to-one. If the function is one-to-one, then
determine the inverse.
a){(2, 5), (3, 6), (4, 8), (8, 7)}
b){(0, 3), (-1, 2), (1, 3)}
c) A Norwegian physiologist has developed a rule for predicting running times
based on the
time to 5 km (5K). An example for one runner is shown here.
| Distance |
Time |
| 1.5K |
4:22 |
| 3K |
9:18 |
| 5K |
16:00 |
| 10K |
33:40 |
Horizontal Line Test
•A function is one-to-one if every horizontal line intersects the graph of the
function at
most once
•The test is used to determine if the graph of a function is one-to-one
Examples: Use the horizontal line test to determine whether each graph is
the graph of a one-to-
one function.
•Given an equation we can determine whether it is
one-to-one and then find its inverse
Finding the Equation of the Inverse
1.Interchange x and y
2. Solve for y
3.Replace y with 
Examples: Decide whether each equation defines a one-to-one function. If
the function is one-
to-one then determine its inverse.
Graphing the Inverse
• Given the graph of f we can determine the graph of f
-1 assuming is one-to-one
• Locate the mirror image of each point
• The line y = x is the “mirror line” or the line of
symmetry
Examples: Use the given graph to graph the inverse.
Section 12.2: Exponential Functions
In an exponential function, the base is a constant and the
exponent is the variable
For a>0, a≠1, and all real numbers x ,
defines
an exponential function with
a base of a
Graph of
1. The graph contains the point (0, 1)
2. When a>1, the graph will rise from left to right. When
0<a<1,the graph will fall
from left to right .In both cases, the graph goes from the second quadrant to
the first.
3. The graph will approach the x-axis, but never touch it. (asymptote)
4.The domain is (−∞ ,∞ )and the range is(0, ∞),
Examples: graph the fol lowing functions .
state the domain and range in interval notation
Property for Solving an Exponential Equation
For a>0 and a≠1, if 
then X=y
Solving an exponential equation
1. Each side must have the same base
2. Simplify the exponents as necessary
3.Set the exponents equal to each other
4.Solve the resulting equation
Examples: Solve each equation
•Exponential functions are often used to model growth and
decay
•The following example illustrates this use
Example: The atmospheric pressure, in millibars, at a given altitude x,
in meters, can be
approximated by the following function
for x between 0 and 10,000
a) According to this function, what is the pressure at ground level?
b) What is the pressure at 5000m?
Section 12.3: Logarithmic Functions
• A logarithmic function is the inverse of an exponential function
• To graph a logarithmic function we simply graph the inverse of the
corresponding
exponential function
Logarithm
For all positive numbers a, with a≠1, and all positive numbers x,
means the same as
A logarithm is an exponent. The expression 
represents the exponent to which the base a
must be raised to obtain
Examples: Use the definition of a logarithm to translate between the
exponential and
logarithmic forms given.
•We use the property of logs to solve logarithmic
equations
•The logarithmic equation becomes an exponential equation
•We then use the methods spoken of previously to solve the resulting equation
Examples: Solve each equation.
Properties of Logarithms
For any positive real number b, with b≠1,
•
and 
• 
is undefined
Examples: Evaluate each logarithm.
Logarithmic Functions
If a and x are positive numbers, with a≠1, then
defines the logarithmic function
with base a
Graph of
1.The graph contains the point (0, 1)
2.When a>1 the graph will rise from left to right, from the fourth quadrant to
the first.
When 0<a<1 the graph will fall from left to right, from the first quadrant to
the fourth.
3.The graph will approach the y-axis, but never tough it. (asymptote)
4.The domain is (0,∞)and the range is (−∞,∞)
Note: To graph the logarithmic function we use the log property to
rewrite the function as an
exponent. Also, recall that logarithmic functions and exponential functions are
inverses of each
other.
Examples: Graph the following functions. State the
domain and the range for each function.
Section 12.4: Properties of Logarithms
•The properties discussed in the section can be used to fully simplify
logarithmic
expressions
•These properties can also be used when solving logarithmic equations
Product Rule for Logarithms
If x, y and b are positive real numbers, where b≠1, then
That is, the logarithm of a product is the sum of the logarithms of the factors
Quotient Rule for Logarithms
If x, y and b are positive real numbers, where b≠1, then
That is, the logarithm of a quotient is the difference between the logarithm of
the numerator and
the logarithm of the denominator
Power Rule for Logarithms
If x and b are positive real numbers, where b≠1, and if r is any real number,
then
That is, the logarithm of a number to a power equals the exponent times the
logarithm of the
number
Special Properties
If b >0 and b≠1, then
Examples: Use the properties of logarithms to rewrite the given
expressions.
Examples: Use the properties of logarithms to
rewrite each expression if possible . Assume that
all variables represent real numbers.
Examples: Tell whether each statement is true or
false. If the statement is false, state why.
Section 12.5: Common and Natural Logarithms
•We can evaluate logarithms using a calculator
•A common logarithm is one with a base of 10
•A natural logarithm is one with a base of e, where e = 2.71828…
•The natural logarithm is denoted as ln
• Calculators can only calculate common and natural logarithms. Logarithms with
bases
other than 10 cannot be directly evaluated using a calculator, a change of base
formula
must be used to convert each logarithm to a common logarithm
Change of Base Rule
If a>0, a≠1, b>0, b≠1 and x>0, then
Examples: Use the change of base formula to rewrite
the given expressions as a common log.
Example: Use the function,
, to determine the percent of women
who returned to work after having a baby in 1990. In the equation x = 0
represents 1980.
Section 12.6: Exponential and Logarithmic Equations;
Further Applications
•We have solve exponential and logarithmic equations in past sections
•General methods for solving exponential and logarithmic equations are presented
in this
section
Properties for Solving Exponential and Logarithmic Equations
For all real numbers b>0, b≠1, and any real numbers x and y
1.If x=y then 
2.If then x=y
3.If x=y and x>0, y>0, then 
4.If x>0, y>0 and ,then x=y
General Method for Solving an Exponential Equation
1.Take logarithms to the same base on both sides
2.Use the power rule of logarithms or the special property
•As a special case, if both sides can be written as exponentials with the same
base, do so, and
set the exponents equal
Solving a Logarithmic Equation
1.Transform the equation so that a single logarithm appears on each side
2.Use the property, if
then x=y
OR
3.Write the equation in exponential form
•If 
then 
Examples: Solve the equation. Give the exact
solution ; no decimal approximations .
Applications of Logarithmic and Exponential Equations|
• Compound Interest for a finite number of periods
•Continuous Compound Interest
Compound Interest Formula for a Finite Number of periods
If a principle of P dollars is deposited at an annual rate of interest r
compounded (paid) n times
per year, then the account will contain
dollars after t years.
Continuous Compound Interest
If a principle of P dollars is deposited at an annual rate of interest r
compounded continuously
for t years, the final amount on deposit is
Examples: Use the compound interest formulas to
find the solution.
1.Find the value of $2000 deposited at 5% compounded annually for 10 years.
2.Find the number of years it will take for $500 deposited
in an account paying 4% interest
compounded semiannually to double.
3.Suppose that $2000 is invested at 5% interest for 10 years.
a) How much will the investment grow to if it is compounded continuously?
b) How long will it take the amount to double?
