MTH61: EXPONENTIAL AND LOGARITHMIC FUNCTIONS

DERIVATIVES  OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS

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Logarithmic functions are the inverses of exponential functions. The inverse of the exponential function y = a^x is x = a^y. The logarithmic function y = log_ax is defined to be equivalent to the exponential equation x = a^y. y = log_ax only under the following conditions: x = a^y, a > 0, and a≠1. It is called the logarithmic function with base a.

Consider what the inverse of the exponential function means: x = a^y. Given a number x and a base a, to what power y must a be raised to equal x? This unknown exponent, y, equals log_ax. So you see a logarithm is nothing more than an exponent. By definition, a^log_ax = x, for every real x > 0.

Below are pictured graphs of the form y = log_ax when a > 1 and when 0 < a < 1. Notice that the domain consists only of the positive real numbers, and that the function always increases as x increases.

The domain of a logarithmic function is real numbers greater than zero, and the range is real numbers. The graph of y = log_ax is symmetrical to the graph of y = a^x with respect to the line y = x. This relationship is true for any function and its inverse.

Integration of exponential and logarithmic functions