Logrithm - Analytic Properties - Inverse Function

Inverse Function

The formula for the logarithm of a power says in particular that for any number x,

In prose, taking the x-th power of b and then the base-b logarithm gives back x. Conversely, given a positive number y, the formula

says that first taking the logarithm and then exponentiating gives back y. Thus, the two possible ways of combining (or composing) logarithms and exponentiation give back the original number. Therefore, the logarithm to base b is the inverse function of f(x) = bx.

Inverse functions are closely related to the original functions. Their graphs correspond to each other upon exchanging the x- and the y-coordinates (or upon reflection at the diagonal line x = y), as shown at the right: a point (t, u = bt) on the graph of f yields a point (u, t = log_bu) on the graph of the logarithm and vice versa. As a consequence, log_b(x) diverges to infinity (gets bigger than any given number) if x grows to infinity, provided that b is greater than one. In that case, log_b(x) is an increasing function. For b < 1, log_b(x) tends to minus infinity instead. When x approaches zero, log_b(x) goes to minus infinity for b > 1 (plus infinity for b < 1, respectively).