Exponential function `y=a^x` has all properties that guarantee existence of inverse function:
These properties guarantee existence of function, that is inverse to exponential. This function is defined on `(0,+oo)` and its range is all number line.
This inverse function is denoted by `y=log_a(x)` (logarithm of x with base a).
So, logarithmic function `y=log_a(x)`, where a>0 and `a!=1` is a function that is inverse to the exponential function `y=a^x`.
Logarithmic function has following properties:
We can obtain graph of the function `y=log_a(x)` from the graph of the function `y=a^x` using transformation of symmetry about line y=x. On figure you can see two cases: graph of logarithmic function when a>1 and 0<a<1.