In the **Product of Logarithm of A and Logarithm of B**, we have seen that . In this post, we will see prove the quotient law of logarithm. We will show that

.

In proving, we will use the connection between the logarithm notation and exponent notation. Recall in the previous post that if and only if .

**Theorem**

.

**Proof**

Let and .

In exponent notation, these are equivalent to and .

Now, .

Getting the logarithm of both sides, we have

which means that

.

But and .

Therefore, . .

This is a good explanation of the quotient law for logarithms. I have been meaning to post a series about logarithms on my site as well.