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)`.logarithmic function
  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.