Introduction to Logarithmic
Functions
Unit 3: Exponential and
Logarithmic Functions
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)
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)
f-1(x)
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}
Introduction to Logarithmic Functions
GRAPHS OF EXPONENTIALS AND ITS INVERSE
•Let us graph the exponential function y = 2x
•Table of values:
Introduction to Logarithmic Functions
GRAPHS OF EXPONENTIALS AND ITS INVERSE
•Let us find the inverse the exponential function y = 2x
•Table of values:
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)
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
y = ax
Interchange x  y
x = ay
•We write these functions as:
x = ay
x = ay
exponent
y = logax
y = logax
exponent
base
Introduction to Logarithmic Functions
FINDING THE INVERSE OF AN EXPONENTIAL
x
y
a
y
x =log
Logarithmic
Inverse
of the
Form
Exponential
Exponential
Function
Function
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
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
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
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
d) log1/39=2
(1/3)2 = 1/9
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
Introduction to Logarithmic Functions
EVALUATING LOGARITHMS
Example 3) Find the value of x for each example:
a) log1/327 = x
(1/3)x = 27
(1/3)x = (1/3)-3
x = -3
b) log5x = 3
53 = x
x = 125
c) logx(1/9) = 2
d) log3x = 0
x2 = (1/9)
x = 1/3
30 = x
x=1
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
Introduction to Logarithmic Functions
COMPLETED PRESENTATION
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