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

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 < 1. Notice that the domain consists only of the positive real numbers, and that the function always increases as x increases.

Figure %: Two graphs of y = logax. On the left, y = log10x, and on the right, y = logx.
The domain of a logarithmic function is real numbers greater than zero, and the range is real numbers. The graph of y = logax is symmetrical to the graph of y = ax with respect to the line y = x. This relationship is true for any function and its inverse.

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.    

alogax = x.    

loga(bc) = logab + logac.    

loga() = logab - logac.    

loga(xd) = d logax    

A natural logarithmic function is a logarithmic function with base e. f (x) = logex = ln x, where x > 0. ln x 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 = . The more famous and useful formula for changing bases is commonly called the Change of Base Formula. It allows the base of a logarithmic function to be changed to any positive real number ≠1. It states that logax = . In this case, a, b, and x are all positive real numbers and a, b≠1.

In the next section, we'll discuss some applications of exponential and logarithmic functions.