Logarithmic Functions

Functions

Basic Conceptions

Cartesian Coordinate System

Domain and Range

Inverse Functions

Even-Odd Symmetry of Functions

Periodicity of Functions

Exponential and Logarithmic Functions

Exponential Functions

Logarithmic Functions
Natural Logarithms

Sets and Intervals

Hyperbolic Functions

Logarithmic Functions

Key Topics Remaining: Natural Logarithms

The operation of taking the logarithm is the inverse operation of exponentiation.
It gives the solution of the equation

ax = b

with respect to x:

x = log a b.

Thus, for any positive numbers a and b ( a ≠ 1 ),

if ax = b then x = log a b, and vice versa.

This means that

and

A logarithmic function

y = log a x

is the inverse function of y = ax :

log a ax = x and

The domain of a logarithmic function is the set of positive real numbers.
The range of a logarithmic function is the set of all real numbers.

Note that the domain of an exponential function is the range of a logarithmic function, and the range of an exponential function is the domain of a logarithmic function.

The function log a x is referred to as log x.

Properties of logarithmic function.

log a x = log a z if and only if x = z.
If a > 1 then the logarithmic functions are monotone increasing functions.
That is, log a x > log a z for x > z.
If 0 < a < 1 then the logarithmic functions are monotone decreasing functions.
That is, log a x < log a z for x > z.
If a > 1 and x → + ∞ then log a x → + ∞.
If a > 1 and x → 0 then log a x → – ∞.
If 0 < a < 1 and x → + ∞ then log a x → – ∞.
If 0 < a < 1 and x → 0 then log a x → + ∞.
A graph of a logarithmic function includes the point ( 1, 0 ).

Transformations of logarithmic functions are based on Logarithmic Identities.