What is a logarithmic function?

A logarithm is an exponent.

Assume 1 ≠ a > 0 and x > 0

Definition of Logarithmic Function with base a:

y = logax ↔ ay = x

ln(ay) = ln x = y ln a

y = ln x/ln a

Definite logax = ln x/ln a

Properties:

1. The domain of f is all positive real numbers.
2. The range of f is all positive real numbers.
3. f(1) = 0
4. f is an increasing function when a > 1 and decreasing if 0 < a < 1.

   From the definition of logarithm, it's obvious that

   f(x) = logax and f(x) = ax are inverse functions.