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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.