# Rewriting as a single log

When we say simplify we really mean to say that we want to use as many of the logarithm properties as we can. Rule 7: Exponent of Log Rule Raising the logarithm of a number by its base equals the number. This is a nice fact to remember on occasion.

We will be looking at this property in detail in a couple of sections. But, they all mean the same.

### Expanding logarithms

Rule 7: Exponent of Log Rule Raising the logarithm of a number by its base equals the number. Properties 3 and 4 leads to a nice relationship between the logarithm and exponential function. When we say simplify we really mean to say that we want to use as many of the logarithm properties as we can. Therefore, we need to have a set of parenthesis there to make sure that this is taken care of correctly. Next, utilize the Product Rule to deal with the plus symbol followed by the Quotient Rule to address the subtraction part. We do, however, have a product inside the logarithm so we can use Property 5 on this logarithm. This means that we can use Property 5 in reverse.

Properties 3 and 4 leads to a nice relationship between the logarithm and exponential function. This is how it looks when you solve it. The second logarithm is as simplified as we can make it. In this direction, Property 7 says that we can move the coefficient of a logarithm up to become a power on the term inside the logarithm.

Example 5 Write each of the following as a single logarithm with a coefficient of 1.

We do, however, have a product inside the logarithm so we can use Property 5 on this logarithm.

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