Exponential and Logarithmic Functions 5

5.4 Logarithmic Functions EXPONENTIAL AND 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) = log b x where x is any positive real number, is called the logarithmic function with base b.

Logarithmic Functions Graph f (x) = log 2 x. Example 1

Logarithmic Functions Solution: Let’s choose some values for x where the corresponding values for log 2 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 Example 1 Log 2 because Figure 5.10 Log 2 1 = 0 because 2 0 = 1 Note that the f(x) axis is a vertical asymptote.

Base-10 logarithms are called common logarithms. Common Logarithms: Base 10

Find x if log x = Example 2

Common Logarithms: Base 10 Solution: If log x = , then changing to exponential form yields = x; use the key to find x: x = Therefore x = rounded to five significant digits. 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 ) 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 log e 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) = e x. log e x = ln x