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5
Exponential and Logarithmic Functions
EXPONENTIAL AND LOGARITHMIC FUNCTIONS
5.4
Logarithmic Functions
Objectives
• Graph logarithmic functions.
• Evaluate common logarithms.
• Evaluate natural logarithms.
Logarithmic Functions
Definition 5.3
If b > 0 and b  1, then the function defined by
f (x) = logb x
where x is any positive real number, is called the
logarithmic function with base b.
Logarithmic Functions
Graph f (x) = log2 x.
Example 1
Logarithmic Functions
Example 1
Solution:
Let’s choose some values for x where the corresponding
values for log2 x are easily determined. (Remember that
logarithms are defined only for the positive real numbers.)
We plot the points determined by the table and connect them
with a smooth curve to produce Figure 5.10.
Log2
1 1
1
3
 3 because 2  3 
2
8
8
Log2 1 = 0 because 20 = 1
Note that the f(x) axis is a vertical asymptote.
Figure 5.10
Common Logarithms: Base 10
• Base-10 logarithms are called common logarithms.
Common Logarithms: Base 10
Find x if log x = 0.2430.
Example 2
Common Logarithms: Base 10
Example 2
Solution:
If log x = 0.2430, then changing to exponential form
yields 100.2430 = x; use the 10 x key to find x:
x = 100.2430  1.749846689
Therefore x = 1.7498 rounded to five significant
digits.
Common Logarithms: Base 10
• The common logarithmic function is
defined by the equation f (x) = log x.
Natural Logarithms — Base e
• In many practical applications of logarithms, the
number e (remember e 2.71828) is used as a
base. Logarithms with a base of e are called
natural logarithms, and the symbol ln x is
commonly used instead of loge x:
loge x = ln x
• The natural logarithmic function is defined by the
equation f(x) = ln x. It is the inverse of the natural
exponential function g(x) = ex.