You go a concert and you want to know how loud it is in decibels. The decibel level of a sound is found by first assigning an intensity I0 to a very soft sound, or the threshold. The decibel level can then be measured with the formula
where
*
I
*
is the intensity of the sound. If the intensity of the concert is 1,000,000,000(I0), what is its decibel level?

### Guidance

You can probably guess that
in
and
in
. But, what is
if
? Until now, we did not have an inverse to an exponential function. But, because we have variables in the exponent, we need a way to get them out of the exponent. Introduce the logarithm. A
**
logarithm
**
is defined as the inverse of an exponential function. It is written
such that
. Therefore, if
(
**
exponential form
**
), then
(
**
logarithmic form
**
).

There are two special logarithms, or logs. One has base 10, and rather that writing
, we just write log. The other is the
**
natural log
**
, the inverse of the natural number. The natural log has base
and is written
This is the only log that is not written using
.

#### Example A

Rewrite in exponential form.

**
Solution:
**
Use the definition above, also called the “key”.

#### Example B

Find:

a)

b)

c)

**
Solution:
**
Using the key, we can rearrange all of these in terms of exponents.

a) .

b) .

c) . There is no solution. A positive number when raised to any power will never be negative.

There are two special logarithms that you may encounter while writing them into exponential form.

The first is , because . The second is because can be any number except 1.

#### Example C

Use your calculator to find the following logarithms. Round your answer to the nearest hundredth.

a)

b)

c)

**
Solution:
**

a) Locate the
**
LN
**
button on your calculator. Depending on the brand, you may have to input the number first. For a TI-83 or 84, press
**
LN
**
, followed by the 7 and
**
ENTER
**
. The answer is 1.95.

b) The
**
LOG
**
button on the calculator is base 10. Press
**
LOG
**
, 35,
**
ENTER
**
. The answer is 1.54.

c) To use the calculator for a base other than 10 or the natural log, you need to use the change of base formula.

**
Change of Base Formula:
**
, such that
and
and
and
.

So, to use this for a calculator, you can use either LN or LOG.

or

In the TI-83 or 84, the keystrokes would be
**
LOG
**
(226)/
**
LOG
**
(5),
**
ENTER
**
.

**
Intro Problem Revisit
**
Plug the given values into the equation
and solve for
*
d
*
.

Therefore, the decibel level of the concert is 90.

### Guided Practice

1. Write in logarithmic form.

2. Evaluate the following expressions without a calculator.

a)

b)

c)

3. Use a calculator to evaluate each expression. Round your answers to the hundredths place.

a)

b)

c)

4. Use the change of base formula to evaluate in a calculator.

#### Answers

1. Using the key, we have: .

2. Change each logarithm into exponential form and solve for .

a) . must be negative because the answer is not a fraction, like the base.

, so . Therefore, .

b) . , therefore, .

c) . First, , so . To make this a fraction, we need to make the power negative. , therefore .

3. Using a calculator, we have:

a) 3.47 b) 2.33 c) 1.81

4. Rewriting using the change of base formula, we have: . Plugging it into a calculator, we get .

### Vocabulary

- Logarithm
- The inverse of an exponential function and is written such that .

- Exponential Form
- , such that is the base and is the exponent.

- Logarithmic Form
- , such that is the base.

- Natural Log
- The inverse of the natural number, , written .

- Change of Base Formula
- Let and be positive numbers, and . Then, . More specifically, and , so that expressions can be evaluated using a calculator.

### Practice

Convert the following exponential equations to logarithmic equations.

Convert the following logarithmic equations to exponential equations.

convert the following logarithmic expressions without a calculator.

Evaluate the following logarithmic expressions using a calculator. You may need to use the Change of Base Formula for some problems.