Algebra II : Logarithms with Exponents

Study concepts, example questions & explanations for Algebra II

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Example Questions

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Example Question #1 : Logarithms With Exponents

In this question we will use the notation  to represent the base 10 or common logarithm, i.e. .

Find  if .

Possible Answers:

Correct answer:

Explanation:

We can use the Property of Equality for Logarithmic Functions to take the logarithm of both sides:

Use the Power Property of Logarithms:

Divide each side by  :

Use a calculator to get:

   

or

Example Question #2 : Logarithms With Exponents

Simplify 

Possible Answers:

Correct answer:

Explanation:

Using Rules of Logarithm recall: 

 

Thus, in this situation we bring the 2 in front and we get our solution. 

Example Question #3 : Logarithms With Exponents

Simplify the following equation.

Possible Answers:

Correct answer:

Explanation:

We can simplify the natural log exponents by using the following rules for naturla log.

Using these rules, we can perform the following steps.

Knowing that the e cancels the exponential natural log, we can cancel the first e.

Distribute the square into the parentheses and calculate.

Remember that a negative exponent is equivalent to a quotient. Write it as a quotient and then you're finished.

Example Question #4 : Logarithms With Exponents

Evaluate the following expression

Possible Answers:

Correct answer:

Explanation:

Since the exponent is inside the parentheses, you must take the square of 1000 before finding the logarithim.  Therefore

because 

Example Question #5 : Logarithms With Exponents

Evaluate the following expression

 

Possible Answers:

Correct answer:

Explanation:

Since the exponent is inside the parentheses, you must determine the value of the exponential expression first.

then you solve the logarithm

   because 

Example Question #6 : Logarithms With Exponents

Evaluate the following for all integers of  and 

 

Possible Answers:

Correct answer:

Explanation:

 gives us the exponent to which  must be raised to yield 

When  is actually raised to that number in the equation given, the answer must be 

Example Question #7 : Logarithms With Exponents

Evaluate the following expression

 

Possible Answers:

Correct answer:

Explanation:

This is a simple exponent of a logarithmic answer.

 because 

Example Question #8 : Logarithms With Exponents

Evaluate the following expression

Possible Answers:

Correct answer:

Explanation:

This is a two step problem.  First find the log base 2 of 16

   because 

then 

Example Question #9 : Logarithms With Exponents

Which of the following equations is valid?

Possible Answers:

none of the other answers are correct

Correct answer:

Explanation:

Since a logarithm answers the question of which exponent to raise the base to receive the number in parentheses, if the number in parentheses is the base raised to an exponent, the exponent must be the answer.

Example Question #10 : Logarithms With Exponents

Rewrite the following logarithmic expression into expanded form (that is, using addition and/or subtraction):

Possible Answers:

Correct answer:

Explanation:

Before we do anything, the exponent of 4 must be moved to the front of the expression, as the rules of logarithms dictate. We end up with . Remember that a product inside of a logarithm can be rewritten as a sum: . Distributing, we get .

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