6.2: Exponential and Logarithmic Functions
Learning Objectives
A student will be able to:
 Understand and use the basic definitions of exponential and logarithmic functions and how they are related algebraically.
 Distinguish between an exponential and logarithmic functions graphically.
A Quick Algebraic Review of Exponential and Logarithmic Functions
Exponential Functions
Recall from algebra that an exponential function is a function that has a constant base and a variable exponent. A function of the form
Logarithmic Functions
Recall from your previous courses in algebra that a logarithm is an exponent. If the base
This is equivalent to the exponential form
For example, the following table shows the logarithmic forms in the first row and the corresponding exponential forms in the second row.
Historically, logarithms with base of
We denote the natural logarithm of
The table below shows this operation.
A Comparison between Logarithmic Functions and Exponential Functions
Looking at the two graphs of exponential functions above, we notice that both pass the horizontal line test. This means that an exponential function is a onetoone function and thus has an inverse. To find a formula for this inverse, we start with the exponential function
Interchanging
Projecting the logarithm to the base
Thus
This implies that the graphs of
Similarly, in the special case when the base
and
The graph below shows this relationship:
Before we move to the calculus of exponential and logarithmic functions, here is a summary of the two important relationships that we have just discussed:
 The function
y=bx is equivalent tox=logby ify>0 andx∈R.  The function
y=ex is equivalent tox=lny ify>0 andx∈R.
You should also recall the following important properties about logarithms:

logbvw=logbv+logbw 
logbvw=logbv−logbw 
logbwn=nlogbw  To express a logarithm with base in terms of the natural logarithm:
logbw=lnwlnb  To express a logarithm with base
b in terms of another basea :logbw=logawlogab
Review Questions
Solve for

6x=1216 
ex=3 
log2z=3 
lnx2=5 
3e−5x=132 
e2x−7ex+10=0 
−4(3)x=−36 
lnx−ln3=2 
y=5log10(22−x) 
y=3e−2x/3
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Date Created:
Feb 23, 2012Last Modified:
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