# Logarithmic Form

How to write equations in logarithmic form to exponential form and vice versa: definition, examples, and their solutions.

## Definition

Logarithm is a way to write

the exponent of a number.

The exponent 0 satisfies 2^{0} = 1.

So 0 = log_{2} 1.

The exponent 1 satisfies 2^{1} = 2.

So 1 = log_{2} 2.

The exponent 2 satisfies 2^{2} = 4.

So 2 = log_{2} 4.

Likewise,

the exponent 'log_{2} 3' is a number

that satisfies 2^{log2 3} = 3.

(log_{2} 3 = 1.585..., 2^{1.585...} = 3)

And the exponent 'log_{2} 5' is a number

that satisfies 2^{log2 5} = 5.

(log_{2} 5 = 2.322..., 2^{2.322...} = 5)

log_{2} 3 is read "log base 2 of 3".

## Example 1

The exponent 4 satisfies 2^{4} = 16.

So 4 = log_{2} 16.

The exponent -2 satisfies 3^{-2} = 1/9.

So -2 = log_{3} (1/9).

The exponent 1/2 satisfies 5^{1/2} = √5.

So 1/2 = log_{5} √5.

## Example 2

The exponent is 2.

So 3^{2} = 9.

The exponent is -5.

So 2^{-5} = 1/32.

The exponent is 2/3.

So 7^{2/3} = ^{3}√49.