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Logarithm and Exponential Graphing Help

By — McGraw-Hill Professional
Updated on Aug 31, 2011

Logarithm and Exponential Graphing

If a > 0 and f(x) = log a x, x > 0, then

Transcendental Functions 6.4 Calculus with Logs and Exponentials to Arbitrary Bases

Using this information, we can sketch the graph of f ( x ) = log a x .

If a > 1 then ln a > 0 so that f′ ( x ) > 0 and f″ ( x ) < 0. The graph of f is exhibited in Fig. 6.6.

Transcendental Functions 6.4 Calculus with Logs and Exponentials to Arbitrary Bases

Fig. 6.6

If 0 < a < 1 then ln a = − ln(1/ a ) < 0 so that f′ ( x ) < 0 and f″ ( x ) > 0. The graph of f is sketched in Fig. 6.7.

Since g ( x ) = a x is the inverse function to f ( x ) = log a x , the graph of g is the reflection in the line y = x of the graph of f (Figs 6.6 and 6.7). See Figs 6.8, 6.9.

Transcendental Functions 6.4 Calculus with Logs and Exponentials to Arbitrary Bases

Fig. 6.7

Transcendental Functions 6.4 Calculus with Logs and Exponentials to Arbitrary Bases

Fig. 6.8

Transcendental Functions 6.4 Calculus with Logs and Exponentials to Arbitrary Bases

Fig. 6.9

Figure 6.10 shows the graphs of log a x for several different values of a > 1.

Figure 6.11 shows the graphs of a x for several different values of a > 1.

You Try It : Sketch the graph of y = 4 x and y = log 4 x on the same set of axes.

Transcendental Functions 6.4 Calculus with Logs and Exponentials to Arbitrary Bases

Fig. 6.10

Transcendental Functions 6.4 Calculus with Logs and Exponentials to Arbitrary Bases

Fig. 6.11

Find practice problems and solutions for these concepts at: Transcendental Functions Practice Test.

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