### All High School Math Resources

## Example Questions

### Example Question #1 : Logarithms

**Possible Answers:**

**Correct answer:**

Most of us don't know what the exponent would be if and unfortunately there is no on a graphing calculator -- only (which stands for ).

Fortunately we can use the base change rule:

Plug in our given values.

### Example Question #1 : Logarithms

Based on the definition of logarithms, what is ?

**Possible Answers:**

2

100

3

10

4

**Correct answer:**

3

For any equation , . Thus, we are trying to determine what power of 10 is 1000. , so our answer is 3.

### Example Question #1 : Logarithms

What is the value of that satisfies the equation ?

**Possible Answers:**

**Correct answer:**

is equivalent to . In this case, you know the value of (the argument of a logarithmic equation) and b (the answer to the logarithmic equation). You must find a solution for the base.

### Example Question #1 : Simplifying Logarithms

Simplify .

**Possible Answers:**

**Correct answer:**

Using properties of logs we get:

### Example Question #1 : Simplifying Logarithms

Simplify the following expression:

**Possible Answers:**

**Correct answer:**

Recall the log rule:

In this particular case, and . Thus, our answer is .

### Example Question #1 : Simplifying Logarithms

Use the properties of logarithms to solve the following equation:

**Possible Answers:**

No real solutions

**Correct answer:**

Since the bases of the logs are the same and the logarithms are added, the arguments can be multiplied together. We then simplify the right side of the equation:

The logarithm can be converted to exponential form:

Factor the equation:

Although there are two solutions to the equation, logarithms cannot be negative. Therefore, the only real solution is .

### Example Question #1 : Simplifying Logarithms

Which of the following represents a simplified form of ?

**Possible Answers:**

**Correct answer:**

The rule for the addition of logarithms is as follows:

.

As an application of this,.

### Example Question #1 : Simplifying Logarithms

Simplify the expression using logarithmic identities.

**Possible Answers:**

The expression cannot be simplified

**Correct answer:**

The logarithm of a fraction is equal to the logarithm of the numerator minus the logarithm of the denominator.

If we encounter two logarithms with the same base, we can likely combine them. In this case, we can use the reverse of the above identity.

### Example Question #1 : Simplifying Logarithms

Evaluate by hand

**Possible Answers:**

Cannot be found by hand

**Correct answer:**

Using the logarithm rules, exponents within logarithms can be removed and simply multiplied by the remaining logarithm. This expression can be simplified as

### Example Question #2 : Logarithms

Solve for

**Possible Answers:**

**Correct answer:**

Use the power reducing theorem:

and