Definition of the logarithm, its properties and graph

Logarithm of a number is the power to which one number must be raised to obtain another.

If the number b to the extent y equals x:

b^y = x

So the logarithm of the number x by reason b is y:

y = log_b(X)

For example:

2⁴ = 16

log₂(16) = 4

Logarithm as inverse function to exponential

logarithmic function y = log_b(x) is the inverse function of the exponential x=b ^y.

So if we calculate the exponential function of the logarithm x (x > 0), it will turn out:

f (f ^-1(x)) = b^logb^(x) = x

Or if we calculate the logarithm of the exponential function х:

f ^-1(f (x)) = log_b(b^x) = x

Natural logarithm (ln)

The natural logarithm is the base logarithm е.

ln (x) = log_e(x)

Number e is a constant that can be defined as a limit:

Or so:

Inverse logarithm

Inverse logarithm (or antilogarithm) of a number n is a number whose base logarithm is a is equal to the number n.

ant log_an = aⁿ

Table of properties of logarithms

Below are the main properties of logarithms in tabular form.