# Algebra II : Logarithms with Exponents

## 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 .

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

Explanation:

Using Rules of Logarithm recall:

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

### Example Question #1 : Logarithms With Exponents

Simplify the following equation.

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

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

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

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

Explanation:

This is a simple exponent of a logarithmic answer.

because

### Example Question #8 : Logarithms With Exponents

Evaluate the following expression

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?

none of the other answers are correct

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):