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

2. Introduction to Logarithmic Functions
GRAPHS OF EXPONENTIALS AND ITS INVERSE
In Grade 11, you were introduced to inverse functions.
Inverse functions is the set of ordered pair obtained by interchanging the x and y values.
f(x)
f-1(x)

3. 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)

4. 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}

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

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

7. 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)

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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) log1/39=2

14. 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

15. 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

16. 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

17. Introduction to Logarithmic Functions
COMPLETED PRESENTATION
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