# Algebra II : Adding and Subtracting Logarithms

## Example Questions

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### Example Question #41 : Adding And Subtracting Logarithms

Combine the logs as one:       Explanation:

Evaluate each term.  Write the property of logs when they are subtracted.   The remaining term is: The answer is: ### Example Question #42 : Adding And Subtracting Logarithms

Solve .      Explanation:

When you add logarithms of the same base, you multiply the terms inside the log: A logarithm with a base that's the same as the term inside it is always equal to .

### Example Question #43 : Adding And Subtracting Logarithms

Solve .      Explanation:

When adding logs with the same base, we multiply the terms inside of them: Now we can expand the log again so that the terms inside of it match the base: ### Example Question #44 : Adding And Subtracting Logarithms

Combine as one log:       Explanation:

According to log rules, whenever we are adding the terms of the logs, we can simply combine the terms as one log by multiplication.  The answer is: ### Example Question #45 : Adding And Subtracting Logarithms

Simplify .      Explanation:

First we can make the coefficient from the left term into an exponent: Next, remember that if we're subtracting logs, we divide the terms inside them: ### Example Question #46 : Adding And Subtracting Logarithms

Simplify       Explanation:

When we add logs, we multiply the terms in them: From here, we multiply them out: ### Example Question #47 : Adding And Subtracting Logarithms

True or false: for all positive values of  False

True

True

Explanation:

By the Product of Logarithms Property, Setting , this becomes "log" refers to the common, or base ten, logarithm, so, by definition, if and only if .

Setting    ,

so and .

The statement is true.

### Example Question #48 : Adding And Subtracting Logarithms

True or false: for all values of .

True

False

False

Explanation:

A statement can be proved to not be true in general if one counterexample can be found. One such counterexample assumes that . The statement becomes or, equivalently, The word "log" indicates a common, or base ten, logarithm, as opposed to a natural, or base , logarithm. By definition, the above statement is true if and only if or .

This is false, so does not hold for . Since the statement fails for one value, it fails in general.

### Example Question #49 : Adding And Subtracting Logarithms

True or false: for all negative values of .

False

True

False

Explanation:

It is true that by the Product of Logarithms Property, .

However, this only holds true if both and are positive. The logarithm of a negative number is undefined, so the expression is undefined. The statement is therefore false.

### Example Question #50 : Adding And Subtracting Logarithms

True or false: for all positive .

False

True

True

Explanation:

By the Change of Base Property of Logarithms, if and  Substituting 7 for and 6 for , the statement becomes the given statement .

The correct choice is "true."

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