# 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:

Also Read:

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 log_{5} 125 = 3 and log_{5} 25 = 2, find log_{5} (125 x 25).

Solution:

Log_{b} cd = log_{b} c + log_{b} d

log_{5} (125 x 25) = log_{5} 125 + log_{5} 25

log_{5} (125 x 25) = 3 + 2

log_{5} (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

Example 2:

Use logs to show that for *b *> 0, b^{0} = 1.

Solution:

Let log_{b} 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: