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 *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 **logarithm** is defined as the inverse of an exponential function. It is written **exponential form**), then **logarithmic form**).

There are two special logarithms, or logs. One has base 10, and rather that writing **natural log**, the inverse of the natural number. The natural log has base

#### Example A

Rewrite

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

The first is

#### 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:**

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

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 *d*.

Therefore, the decibel level of the concert is 90.

### Guided Practice

1. Write

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

#### Answers

1. Using the key, we have:

2. Change each logarithm into exponential form and solve for

a)

\begin{align*}2^4=16\end{align*}, so \begin{align*}\left(\frac{1}{2}\right)^{-4}=16\end{align*}. Therefore, \begin{align*}\log_{\frac{1}{2}}16=-4\end{align*}.

b) \begin{align*}\log100 \rightarrow 10^x=100\end{align*}. \begin{align*}x=2\end{align*}, therefore, \begin{align*}\log100=2\end{align*}.

c) \begin{align*}\log_{64} \frac{1}{8} \rightarrow 64^x = \frac{1}{8}\end{align*}. First, \begin{align*}\sqrt{64}=8\end{align*}, so \begin{align*}64^{\frac{1}{2}}=8\end{align*}. To make this a fraction, we need to make the power negative. \begin{align*}64^{-\frac{1}{2}}=\frac{1}{8}\end{align*}, therefore \begin{align*}\log_{64} \frac{1}{8}=-\frac{1}{2}\end{align*}.

3. Using a calculator, we have:

a) 3.47 b) 2.33 c) 1.81

4. Rewriting \begin{align*}\log_8 \frac{7}{9}\end{align*} using the change of base formula, we have: \begin{align*}\frac{\log \frac{7}{9}}{\log 8}\end{align*}. Plugging it into a calculator, we get \begin{align*}\frac{\log \left(\frac{7}{9}\right)}{\log 8} \approx -0.12\end{align*}.

### Explore More

Convert the following exponential equations to logarithmic equations.

- \begin{align*}3^x=5\end{align*}
- \begin{align*}a^x=b\end{align*}
- \begin{align*}4(5^x)=10\end{align*}

Convert the following logarithmic equations to exponential equations.

- \begin{align*}\log_2 32=x\end{align*}
- \begin{align*}\log_{\frac{1}{3}}x=-2\end{align*}
- \begin{align*}\log_a y=b\end{align*}

convert the following logarithmic expressions without a calculator.

- \begin{align*}\log_5 25\end{align*}
- \begin{align*}\log_{\frac{1}{3}} 27\end{align*}
- \begin{align*}\log \frac{1}{10}\end{align*}
- \begin{align*}\log_2 64\end{align*}

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

- \begin{align*}\log 72\end{align*}
- \begin{align*}\ln 8\end{align*}
- \begin{align*}\log_2 12\end{align*}
- \begin{align*}\log_3 9\end{align*}
- \begin{align*}\log_{11} 32\end{align*}