# Precalculus : Exponential and Logarithmic Functions

## Example Questions

### Example Question #1 : Exponential And Logarithmic Functions

You are given that  and

Which of the following is equal to  ?

Explanation:

Since  and , it follows that  and

### Example Question #1 : Logarithmic Functions

Solve.

Explanation:

First we condense the LHS to  using the properties of logarithms.  Then we can eliminate the natural log on the LHS by raising both sides of the equation as exponents with base . This leaves us with  since .  Now we do some simple Algebra and obtain our answer.

### Example Question #1 : Exponential And Logarithmic Functions

Solve.

Explanation:

First take the logarithm of both sides.  This yields . Now using the properties of logarithms, we can bring the  down in front of . Since . We are left with .  Dividing each side by  we find .

### Example Question #4 : Exponential And Logarithmic Functions

Evalute the equation . Use a calculator to approximate the answer to three decimal places.

Explanation:

First, we rewrite the equation in logarithmic form to obtain . Simplifying, we get

Simplify

Explanation:

### Example Question #1 : Exponential And Logarithmic Functions

Solve the following equation for :

Explanation:

Logs must be used to solve this problem. Follow the steps below:

### Example Question #1 : Exponential And Logarithmic Functions

Solve.

Explanation:

First we condense the LHS to . Now we can eliminate a log of base  by making both sides of the equation exponents of base . This leaves us with . Moving  to the LHS we obtain a simple quadratic which we solve to obtain  and  for solutions. Finally, we eliminate  as logarithms are undefined for negative numbers.

### Example Question #1 : Logarithmic Functions

Find the domain of the function   .

Explanation:

The logarithmic function is undefined when the inputs are negative or 0. Therefore the inputs of the logarithmic function must be positive. This means that the quantity  must be positive. After setting up the appropriate inequality, we have,

Therefore the domain of the function    is the interval .

### Example Question #1 : Exponential And Logarithmic Functions

Use the logarithmic base change formula to convert  to a quotient of logarithms with a base of .

Explanation:

Here we employ the use of the logarithm base change formula

.

Therefore we get,

### Example Question #1 : Logarithmic Functions

Find the domain of the function .

Explanation:

The logarithmic function is undefined when the inputs are negative or 0. Therefore the inputs of the logarithmic function must be positive. This means that the quantity  must be positive. However, this quantity is always positive because the value under the radical cannot be negative. So we are only constrained by:

Therefore the domain of the function  is the interval .