# Rewrite as a single logarithm and simplify calculator

Exponential equations in which the unknown occurs just once To solve this type of equation, follow these steps: Invert the operations that were applied to the exponential in the reverse order in which they were applied. Properties 3 and 4 leads to a nice relationship between the logarithm and exponential function. This is a nice fact to remember on occasion. We will be looking at this property in detail in a couple of sections.

We will just need to be careful with these properties and make sure to use them correctly. Also, note that there are no rules on how to break up the logarithm of the sum or difference of two terms.

Note that all of the properties given to this point are valid for both the common and natural logarithms. Example 4 Simplify each of the following logarithms.

When we say simplify we really mean to say that we want to use as many of the logarithm properties as we can. In order to use Property 7 the whole term in the logarithm needs to be raised to the power.

We do, however, have a product inside the logarithm so we can use Property 5 on this logarithm. In these cases it is almost always best to deal with the quotient before dealing with the product. Here is the first step in this part. Therefore, we need to have a set of parenthesis there to make sure that this is taken care of correctly.

The second logarithm is as simplified as we can make it. Also, we can only deal with exponents if the term as a whole is raised to the exponent. It needs to be the whole term squared, as in the first logarithm. Here is the final answer for this problem.

This next set of examples is probably more important than the previous set. We will be doing this kind of logarithm work in a couple of sections. Example 5 Write each of the following as a single logarithm with a coefficient of 1. We will have expressions that look like the right side of the property and use the property to write it so it looks like the left side of the property.

This will use Property 7 in reverse. 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. Here is that step for this part. This means that we can use Property 5 in reverse.

Here is the answer for this part. When using Property 6 in reverse remember that the term from the logarithm that is subtracted off goes in the denominator of the quotient. Here is the answer to this part. The reason for this will be apparent in the next step.is and in to a was not you i of it the be he his but for are this that by on at they with which she or from had we will have an what been one if would who has her.

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Algebra - Logarithm Functions