Algebra II : Solving and Graphing Logarithms

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #31 : Solving Logarithms

Expand the log:  

Possible Answers:

Correct answer:

Explanation:

Rewrite the logarithm using the quotient property.

Rewrite the log.

Use the power property to pull down the power.

Simplify the log terms.

The answer is:  

Example Question #32 : Solving Logarithms

Solve for :

Possible Answers:

None of these

Correct answer:

Explanation:

Using definition of logarithms:

Example Question #33 : Solving Logarithms

Solve for :

Possible Answers:

None of these

Correct answer:

Explanation:

Using definition of logarithms

Example Question #34 : Solving Logarithms

Solve the following equation:

Possible Answers:

Correct answer:

Explanation:

Recall the definition of a logarithm. For a logarithm in any base 

denotes

.

In other words, the value of a logarithm is simply an exponent, and it is defined at  whenever the base can be raised to a sufficient power to yield . Hence, the equation

may be rewritten as

 .

Raising  to the th power yields , and so

.

Example Question #35 : Solving Logarithms

Solve the following equation:

Possible Answers:

Correct answer:

Explanation:

According to the rule for multiplying logarithms,  for any constant value . In other words, an exponent on the quantity inside a logarithm can be moved outside the logarithm as a multiplier, and vice versa. Hence,

.

Since the natural logarithm has  as its base, the definition of logarithms applies in this case:  whenever the base of  is equal to .

.

This implies that  and . However, substituting the solution  into the original equation yields the following expression:

which is not defined, since logarithms are undefined for any value of .

Hence, we eliminate the extraneous solution  and submit that the correct solution to this equation is .

 

Example Question #36 : Solving Logarithms

Solve for :   

Possible Answers:

Correct answer:

Explanation:

Use the following interchangeable properties to solve for the unknown variable:

For the equation , this would be equivalent to:  

Divide by two on both sides and convert the negative exponent to a fraction.

The answer is:  

Example Question #37 : Solving Logarithms

Solve the log:  

Possible Answers:

Correct answer:

Explanation:

In order to simplify this log, we will need to convert the nine to base three.

Rewrite the term inside the log.

Rewrite this as a fraction.

According to log rules, we can pull the power of the inner quantity inside the log out as the coefficient.

The answer is:  

Example Question #38 : Solving Logarithms

Evaluate:  

Possible Answers:

Correct answer:

Explanation:

The expression can be rewritten as a fraction.

The log based 100 can be rewritten as ten squared.  

The answer is:  

Example Question #39 : Solving Logarithms

Solve the logs:  

Possible Answers:

Correct answer:

Explanation:

For logs that are added, the inner quantities can be multiplied with each other to be combined as a single log.

For logs that are subtracted, the inner quantities would each be divided instead.

Rewrite the expression:

The answer is:  

Example Question #40 : Solving Logarithms

Evaluate:  

Possible Answers:

Correct answer:

Explanation:

Recall that log has a base of ten.

The equation can be rewritten as:

Add 1 on both sides.

To eliminate the log based 10 on the left, we will need to raise both terms as exponents of base 10.

The answer is:  

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