# Algebra II : Multiplying and Dividing Logarithms

## Example Questions

### Example Question #1 : Multiplying And Dividing Logarithms

Rewrite the following logarithmic expression in expanded form (i.e. as a sum and/or difference):

Explanation:

By logarithmic properties:

;

Combining these three terms gives the correct answer:

### Example Question #1 : Simplifying Logarithms

Which of the following is equivalent to

Explanation:

Recall that log implies base  if not indicated.Then, we break up . Thus, we have .

Our log rules indicate that

.

So we are really interested in,

.

Since we are interested in log base , we can solve  without a calculator.

We know that , and thus by the definition of log we have that .

Therefore, we have

### Example Question #3 : Multiplying And Dividing Logarithms

Find the value of the Logarithmic Expression.

Explanation:

Use the change of base formula to solve this equation.

### Example Question #4 : Multiplying And Dividing Logarithms

What is another way of expressing the following?

Explanation:

Use the rule

### Example Question #5 : Multiplying And Dividing Logarithms

Expand this logarithm:

Explanation:

In order to solve this problem you must understand the product property of logarithms  and the power property of logarithms . Note that these apply to logs of all bases not just base 10.

log of multiple terms is the log of each individual one:

now use the power property to move the exponent over:

### Example Question #6 : Multiplying And Dividing Logarithms

Which of the following is equivalent to ?

Explanation:

We can rewrite the terms of the inner quantity.  Change the negative exponent into a fraction.

This means that:

Split up these logarithms by addition.

According to the log rules, the powers can be transferred in front of the logs as coefficients.

### Example Question #7 : Multiplying And Dividing Logarithms

Many textbooks use the following convention for logarithms:

Solve: