3.2 Logarithmic Functions and Their Graphs

In chapters previous, we learned that if a function is one-to-one, then it must have an inverse function. This also applies to exponential functions. This concept flows into the definition of what a logarithmic function is.

Defintion:

In  where  and  there is only one input for every output, making it one-to-one. Every one-to-one function has an inverse. A logarithmic function is the inverse of an exponential function. The inverse of an exponential function is the logarithmic function . for a logarithmic function to occur.

Since the exponential function had a base of , so must the logarithm. With a logarithm, you are trying to find what power the base must be raised to so that you can get . With this in mind, you are now solving for . The logarithm is of  with a base of  solving for .

Examples:
  to 

 to 

to 


With your study of logarithms, you will come into contact with many different kinds of logs. The common log and natural log are two that you will stumble upon frequently.

Common log: 

Natural log:

Graph:

Domain: 
Range:
x-Intercepts:
y-Intercepts: none
Vertical Asymptote: 
Horizontal Asymptote: none

Transformations:






Change in  causes a vertical stretch or compression ( if negative, there will be a reflection across the -axis)
Change in  causes the rate of growth/decay to change
Change in  causes a horizontal stretch or compression ( if negative, there will be a reflection across the -axis
Change in  causes a shift along the  axis
Change in  causes a shift along the  axis

Graphical Representation of Inverse:

If you still need help on logarithmic functions and their graphs, you can go to the following sites listed below.
http://www.purplemath.com/modules/graphlog.htm
http://www.regentsprep.org/regents/math/algtrig/atp8b/logfunction.htm
https://www.khanacademy.org/math/algebra2/exponential-and-logarithmic-functions/graphs-of-logarithmic-functions/v/graphing-logarithmic-functions