# LOGARITHMIC RELATIONSHIPS

Because a logarithm is an exponent, the rules for exponents can be used to derive the rules for logarithms. Just as the rules for exponents only apply to powers with like bases, the rules for logarithms will apply to logarithms with the same base.

Basic Properties of Logarithms

if 0 < b < 1 or b > 1:

For example:

Logarithms of Products

If 0 < b < 1 or b > 1:

Therefore:

The log of a product is the sum of the logs of the factors of the product.

For example:

Example 1:

If log5 125 = 3 and log5 25 = 2, find log5 (125 x 25).

Solution:

Logb cd = logb c + logb d

log5 (125 x 25) = log5 125 + log5 25

log5 (125 x 25) = 3 + 2

log5 (125 x 25) = 5.

Logarithms of Quotients

if 0 < b < 1 or b > 1:

Therefore

The log of a quotient is the log of the dividend minus the log of the divisor.

For example:

Do we get the same answer if the quotient is simplified?

This is true because $3^{2-4}=3^-2=\frac{1}{9}$

Example 2:

Use logs to show that for b > 0, b0 = 1.

Solution:

Let logb c = a, then

Also Read : Logarithmic form of an exponential equation

Logarithms of Powers

If 0 < b < 1 or b > 1:

Therefore

The log of a power is the exponent times the log of the base.

For example:

×