If x = b^{y} where both b > 0, x > 0, then we define y = log_{b} x, which is read as
"y is the log to the base b of x". This means that, log_{b} x or y is the number to which b has to be raised exponentially in order to equal x. So a log is simply the inverse function of an exponential function. Therefore, the domain of any logarithmic function is the set of positive numbers since the range of any exponential function is the set of positive numbers.
Example 20
Let us find the log of 100 to the base 10
As 100 = 10^{2}
log_{10 }100 = 2
Similarly,
The exponential function and logarithmic function are inverse functions of each other, since
y = log_{b} x if and only if x = b^{y}.
If y = log_{10} x = f(x), then x = 10^{y} and vice-versa.
If x = e^{y} then y = ln x and vice versa.
The shape of the logarithmic function y = ln x is graphically illustrated below.