You are watching: A logarithmic function is the inverse of an exponential function
Consider what the inverse of the exponential function means: x = ay. Given a number x and a base a, to what power y must a be raised to equal x? This unknown exponent, y, equals logax. So you see a logarithm is nothing more than an exponent. By definition, alogax = x, for every real x > 0.
Below are pictured graphs of the form y = logax when a > 1 and when 0 a . Notice that the domain consists only of the positive real numbers, and that the function always increases as x increases.
Here are some useful properties of logarithms, which all follow from identities involving exponents and the definition of the logarithm. Remember a > 0, and x > 0.
|loga1 = 0.|
|logaa = 1.|
|loga(ax) = x.|
) = logab - logac.
A natural logarithmic function is a logarithmic function with base e. f (x) = logex = lnx, where x > 0. lnx is just a new form of notation for logarithms with base e. Most calculators have buttons labeled "log" and "ln". The "log" button assumes the base is ten, and the "ln" button, of course, lets the base equal e. The logarithmic function with base 10 is sometimes called the common logarithmic function. It is used widely because our numbering system has base ten. Natural logarithms are seen more often in calculus.
Two formulas exist which allow the base of a logarithmic function to be changed. The first one states this: logab =
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In the next section, we"ll discuss some applications of exponential and logarithmic functions.