# Algebra II : Adding and Subtracting Logarithms

## Example Questions

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

Simplify the following logarithmic expression:

Explanation:

Each term can be simplified as follows:

### Example Question #5 : Simplifying 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:

Explanation:

Use the sum/product rule to combine the first 2 terms:

Use the difference/quotient rule to combine the remaining terms:

### Example Question #3 : Adding And Subtracting Logarithms

Expand the following logarithmic expression into a list of sums or subtractions of logarithms:

Explanation:

One important property of logarithms is that multiplication inside the logarithm is the same thing as addition outside of it. In the same way division is "the same" as subtraction in logarithms. So our expression is the same as

But also, exponents can be moved outside in the same way.  is basically , so . This can be reduced even further to our final answer:

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

What is the value of ?

Explanation:

Remember the rules of logarithms:

This means we can simplify it as follows:

The logarithm of anything with the same base is always , so the correct answer is .

### Example Question #5 : Adding And Subtracting Logarithms

Which of the following is another way to express

?

Explanation:

Use the rule

therefore

### Example Question #6 : Adding And Subtracting Logarithms

Which is another way of expressing

?

Explanation:

Use the rule:

therefore

### Example Question #7 : Adding And Subtracting Logarithms

Simplify

Explanation:

This problem can be solved using the properties of logs. When two logs are being subtracted from each other, it is the same thing as dividing two logs together. Remember that to use this rule, the logs must have the same base in this case .

### Example Question #8 : Adding And Subtracting Logarithms

Condense 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.

first move the constants in front of the logarithmic functions to their proper place using the power rule.

next factor out the logarithmic equation:

change the fractional exponent to a radical

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