Let's start by recalling that we call an exponential function every function of the form f(x)=a^x where a> 0, a≠1, and x is any real number. Each exponential function is one-to-one, which means that no horizontal line intersects the graph more than once. In other words, each exponential function passes the Horizontal Line Test.
Therefore, each exponential function must have an inverse function. This inverse function is called the logarithmic function with base a.
Definition of Logarithmic Function
For x > 0, a> 0, and a≠1, y = log_a x if and only if x=a^y. The function given by f(x) = log_ax is called the logarithmic function with base a.
We can now complete the sentence.
The inverse function of the exponential function f(x)=a^x is called the logarithmic function with base a.
