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Graphing Logarithmic Functions

Let us study the graph of y = lnx. Here are a few values of y and dy/dx.

X

¼

½

1

2

4

y = ln x

-1.4

-0.7

0

0.7

1.4

dy/dx = 1/x

4

2

1

½

¼

The limits as x → 0+ and x → ∞ (see Example 4, Section 8.2) are:

08_exp-log_functions-235.gif

From the sign of dy/dx and d2y/dx2 we get the following information.

 

08_exp-log_functions-236.gif increasing
concave downward.

We use this information to draw the curve in Figure 8.5.2.

08_exp-log_functions-237.gif

Figure 8.5.2

There are two bases for logarithms which are especially useful for different purposes, base 10 and base e. The student should be careful not to confuse the two.

Table 8.5.1

Name

Common Logarithms

Natural Logarithms

base

base 10

base e

symbols

log10x, log x

logex, ln x

use

numerical computation

derivatives and integrals

To pass back and forth between common and natural logarithm we need the constants log10 e ~ 0.4343, In 10 ~ 2.3026.

Then

08_exp-log_functions-238.gif

and

08_exp-log_functions-239.gif

Warning: Do not make the mistake of using common logarithms instead of natural logarithms in differentiating and integrating.


Last Update: 2006-11-16