# Algebra II : Simplifying Logarithms

## Example Questions

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### 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:

Explanation:

Remembering the rules for logarithms, we know that .

This tells us that .

This becomes , which is .

### Example Question #1 : Adding And Subtracting Logarithms

Simplify the following logarithmic expression:

Explanation:

Each term can be simplified as follows:

Combining these gives the answer:

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

Simplify the expression using logarithmic identities.

The expression cannot be simplified

Explanation:

The logarithm of a fraction is equal to the logarithm of the numerator minus the logarithm of the denominator.

If we encounter two logarithms with the same base, we can likely combine them. In this case, we can use the reverse of the above identity.

### Example Question #2 : Adding And Subtracting Logarithms

Use logarithmic properties to simplify this expression: