Exponential and Logarithmic Functions

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5.4Logarithmic Functions

EXPONENTIAL AND LOGARITHMIC FUNCTIONS

Objectives• Graph logarithmic functions.• Evaluate common logarithms.• Evaluate natural logarithms.

Logarithmic Functions

Definition 5.3If 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

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.

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Example 1

Log2 because

Figure 5.10

Log2 1 = 0 because 20 = 1

Note that the f(x) axis is a vertical asymptote.

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• Base-10 logarithms are called common logarithms.

Common Logarithms: Base 10

Common Logarithms: Base 10

Find x if log x = 0.2430.

Example 2

Common Logarithms: Base 10

Solution:If log x = 0.2430, then changing to exponential form

yields 100.2430 = x; use the key to find x:

x = 100.2430 1.749846689

Therefore x = 1.7498 rounded to five significant

digits.

10x

Example 2

• The common logarithmic function is defined by the equation f (x) = log x.

Common Logarithms: Base 10

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:

• 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.

loge x = ln x