Algebra II : Adding and Subtracting Logarithms

Study concepts, example questions & explanations for Algebra II

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Example Questions

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Example Question #41 : Adding And Subtracting Logarithms

Combine the logs as one:  

Possible Answers:

Correct answer:

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 .

Possible Answers:

Correct answer:

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 .

Possible Answers:

Correct answer:

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:  

Possible Answers:

Correct answer:

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 .

Possible Answers:

Correct answer:

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 

Possible Answers:

Correct answer:

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 

Possible Answers:

False

True

Correct answer:

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 .

Possible Answers:

True

False

Correct answer:

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 .

Possible Answers:

False

True

Correct answer:

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 .

Possible Answers:

False

True

Correct answer:

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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