The hypotenuse of a right triangle has a length of . How long is the triangle's hypotenuse?

### Guidance

The last property of logs is the
**
Power Property
**
.

Using the definition of a log, we have . Now, raise both sides to the power.

Let’s convert this back to a log with base , . Substituting for , we have .

Therefore, the Power Property says that if there is an exponent within a logarithm, we can pull it out in front of the logarithm.

#### Example A

Expand .

**
Solution:
**
To expand this log, we need to use the Product Property and the Power Property.

#### Example B

Expand .

**
Solution:
**
We will need to use all three properties to expand this example. Because the expression within the natural log is in parenthesis, start with moving the
power to the front of the log.

Depending on how your teacher would like your answer, you can evaluate , making the final answer .

#### Example C

Condense .

**
Solution:
**
This is the opposite of the previous two examples. Start with the Power Property.

Now, start changing things to division and multiplication within one log.

Lastly, combine like terms.

**
Intro Problem Revisit
**
We can rewrite
and
and solve.

Therefore, the triangle's hypotenuse is 24 units long.

### Guided Practice

Expand the following logarithmic expressions.

1.

2.

3.

4. Condense into one log: .

#### Answers

1. The only thing to do here is apply the Power Property: .

2. Let’s start with using the Quotient Property.

Now, apply the Product Property, followed by the Power Property.

Simplify and solve for . Also, notice that we put parenthesis around the second log once it was expanded to ensure that the would also be subtracted (because it was in the denominator of the original expression).

3. For this problem, you will need to apply the Power Property twice.

**
Important Note:
**
You can write this particular log several different ways. Equivalent logs are:
and
. Because of these properties, there are several different ways to write one logarithm.

4. To condense this expression into one log, you will need to use all three properties.

**
Important Note:
**
If the problem was
, then the answer would have been
. But, because there are no parentheses, the
is in the numerator.

### Vocabulary

- Power Property
- As long as , then .

### Practice

Expand the following logarithmic expressions.

Condense the following logarithmic expressions.