1. Introduction to Logarithmic Functions
Unit 3: Exponential and Logarithmic Functions

2. Introduction to Logarithmic Functions
GOALS OF THE LESSON
Understand what “log” means
Understand how the graph of a logarithmic function relates to an exponential function
Understand how to change expressions between logarithmic and exponential form
Solve for a missing variable, whether it is a base, exponent, or value within the logarithm
Know how to use the “LOG” button on your calculator

3. Introduction to Logarithmic Functions
GRAPHS OF EXPONENTIALS AND ITS INVERSE
You have seen inverse functions before.
Inverse functions is the set of ordered pair obtained by interchanging the x and y values.
f(x)
f-1(x)

4. Introduction to Logarithmic Functions
GRAPHS OF EXPONENTIALS AND ITS INVERSE
Inverse functions can be created graphically by a reflection on the y = x axis.
f(x)
y = x
f-1(x)

5. Introduction to Logarithmic Functions
GRAPHS OF EXPONENTIALS AND ITS INVERSE
A logarithmic function is the inverse of an exponential function
Exponential functions have the following characteristics:
Domain: {x є R}
Range: {y > 0}

6. Introduction to Logarithmic Functions
GRAPHS OF EXPONENTIALS AND ITS INVERSE
Let us graph the exponential function y = 2x
Table of values:

7. Introduction to Logarithmic Functions
GRAPHS OF EXPONENTIALS AND ITS INVERSE
Let us find the inverse the exponential function y = 2x
Table of values:

8. Introduction to Logarithmic Functions
GRAPHS OF EXPONENTIALS AND ITS INVERSE
When we add the function f(x) = 2x to this graph, it is evident that the inverse is a reflection on the y = x axis
f(x)
f-1(x)
f(x)
f-1(x)

9. Introduction to Logarithmic Functions
FINDING THE INVERSE OF AN EXPONENTIAL
Next, you will find the inverse of an exponential algebraically
Write the process in your notes
base
exponent
y = ax
x = ay
Interchange x  y
x = ay
We write these functions as:
x = ay
y = logax
y = logax
base
exponent

10. Introduction to Logarithmic Functions
FINDING THE INVERSE OF AN EXPONENTIAL x y y x
log = a
Inverse of the Exponential Function
Logarithmic Form
Exponential Function

11. Introduction to Logarithmic Functions
CHANGING FORMS
Example 1) Write the following into logarithmic form:
a) 33 = 27
b) 45 = 256
c) 27 = 128
d) (1/3)x=27
ANSWERS

12. Introduction to Logarithmic Functions
CHANGING FORMS
Example 1) Write the following into logarithmic form:
a) 33 = 27
log327=3
b) 45 = 256
log4256=5
c) 27 = 128
log2128=7
d) (1/3)x=27
log1/327=x

13. Introduction to Logarithmic Functions
CHANGING FORMS
Example 2) Write the following into exponential form:
a) log264=6
b) log255=1/2
c) log81=0
d) log1/39=2
ANSWERS

14. Introduction to Logarithmic Functions
CHANGING FORMS
Example 2) Write the following into exponential form:
a) log264=6
26 = 64
b) log255=1/2
251/2 = 5
c) log81=0
80 = 1
(1/3)2 = 1/9
d) log1/31/9=2

15. Introduction to Logarithmic Functions
EVALUATING LOGARITHMS
Example 3) Find the value of x for each example:
a) log1/327 = x
b) log5x = 3
c) logx(1/9) = 2
d) log3x = 0
ANSWERS

16. Introduction to Logarithmic Functions
EVALUATING LOGARITHMS
Example 3) Find the value of x for each example:
a) log1/327 = x
b) log5x = 3
(1/3)x = 27
(1/3)x = (1/3)-3
x = -3
53 = x
x = 125
c) logx(1/9) = 2
d) log3x = 0
x2 = (1/9)
x = 1/3
30 = x
x = 1

17. Introduction to Logarithmic Functions
BASE 10 LOGS
Scientific calculators can perform logarithmic operations. Your calculator has a LOG button.
This button represents logarithms in BASE 10 or log10
Example 4) Use your calculator to find the value of each of the following:
a) log101000
b) log 50
c) log -1000
= Out of Domain
= 3
= 1.699