# Logarithmic Function

Exponential function y=a^x has all properties that guarantee existence of inverse function:

1. Domain is all number line.
2. Range is interval (0,+oo).
3. Function is increasing when a>0 and decreasing when 0<a<1.

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:

1. Domain is interval (0,+oo).
2. Range is all number line.
3. Function is neither even, nor odd.
4. Function is increasing on interval (0,+oo) when a>1, and decreasing on (0,+oo) when 0<a<1.
5. y-axis is a vertical asymptote of the graph (if a>1, then y->-oo as x->0; if 0<a<1 then y->+oo as x->0).

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.