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# 8.5: Defining Logarithms

Difficulty Level: At Grade Created by: CK-12
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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 d=10logII0\begin{align*}d = 10 \cdot \log \frac{I}{I0}\end{align*} 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 x=3\begin{align*}x=3\end{align*} in 2x=8\begin{align*}2^x=8\end{align*} and x=4\begin{align*}x=4\end{align*} in 2x=16\begin{align*}2^x=16\end{align*}. But, what is x\begin{align*}x\end{align*} if 2x=12\begin{align*}2^x=12\end{align*}? 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 logba=x\begin{align*}\log_b a=x\end{align*} such that bx=a\begin{align*}b^x=a\end{align*}. Therefore, if 52=25\begin{align*}5^2=25\end{align*} (exponential form), then log525=2\begin{align*}\log_5 25=2\end{align*} (logarithmic form).

There are two special logarithms, or logs. One has base 10, and rather that writing log10\begin{align*}\log_{10}\end{align*}, we just write log. The other is the natural log, the inverse of the natural number. The natural log has base e\begin{align*}e\end{align*} and is written ln\begin{align*}\ln\end{align*} This is the only log that is not written using log\begin{align*}\log\end{align*}.

#### Example A

Rewrite log327=3\begin{align*}\log_3 27=3\end{align*} in exponential form.

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

logbalog327=xbx=a=333=27\begin{align*}\log_b a &= x \leftrightarrow b^x=a \\ \log_3 27 &= 3 \leftrightarrow 3^3=27 \end{align*}

#### Example B

Find:

a) log1000\begin{align*}\log 1000\end{align*}

b) log7149\begin{align*}\log_7 \frac{1}{49}\end{align*}

c) log12(8)\begin{align*}\log_{\frac{1}{2}}(-8)\end{align*}

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

a) log1000=x10x=1000,x=3\begin{align*}\log 1000=x \Rightarrow 10^x=1000, x=3\end{align*}.

b) log7149=x7x=149,x=2\begin{align*}\log_7 \frac{1}{49}=x \Rightarrow 7^x=\frac{1}{49}, x=-2\end{align*}.

c) \begin{align*}\log_{\frac{1}{2}}(-8)=x \Rightarrow \left(\frac{1}{2}\right)^x =-8\end{align*}. 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 \begin{align*}\log_b 1=0\end{align*}, because \begin{align*}b^0=1\end{align*}. The second is \begin{align*}\log_b b=1\end{align*} because \begin{align*}b^1=b \cdot b\end{align*} can be any number except 1.

#### Example C

a) \begin{align*}\ln7\end{align*}

b) \begin{align*}\log 35\end{align*}

c) \begin{align*}\log_5 226\end{align*}

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: \begin{align*}\log_a x=\frac{\log_b x}{\log_b a}\end{align*}, such that \begin{align*}x, a,\end{align*} and \begin{align*}b>0\end{align*} and \begin{align*}a\end{align*} and \begin{align*}b \ne 1\end{align*}.

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

\begin{align*}\log_5 226=\frac{\log 226}{\log 5}\end{align*} or \begin{align*}\frac{\ln 226}{\ln 5} \approx 3.37\end{align*}

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 \begin{align*}d = 10 \cdot \log \frac{I}{I0}\end{align*} and solve for d.

\begin{align*}d = 10 \cdot \log \frac{1,000,000,000 (I0)}{I0}\\ d = 10 \cdot \log 1,000,000,000\\ d = 10 \cdot 9 = 90\end{align*}

Therefore, the decibel level of the concert is 90.

### Guided Practice

1. Write \begin{align*}6^2=36\end{align*} in logarithmic form.

2. Evaluate the following expressions without a calculator.

a) \begin{align*}\log_{\frac{1}{2}} 16\end{align*}

b) \begin{align*}\log 100\end{align*}

c) \begin{align*}\log_{64} \frac{1}{8}\end{align*}

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

a) \begin{align*}\ln 32\end{align*}

b) \begin{align*}\log_7 94\end{align*}

c) \begin{align*}\log 65\end{align*}

4. Use the change of base formula to evaluate \begin{align*}\log_8 \frac{7}{9}\end{align*} in a calculator.

1. Using the key, we have: \begin{align*}6^2=36 \rightarrow \log_6 36=2\end{align*}.

2. Change each logarithm into exponential form and solve for \begin{align*}x\end{align*}.

a) \begin{align*}\log_{\frac{1}{2}} 16 \rightarrow \left(\frac{1}{2}\right)^x=16\end{align*}. \begin{align*}x\end{align*} must be negative because the answer is not a fraction, like the base.

\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*}.

### Vocabulary

Logarithm
The inverse of an exponential function and is written \begin{align*}\log_b a=x\end{align*} such that \begin{align*}b^x=a\end{align*}.
Exponential Form
\begin{align*}b^x=a\end{align*}, such that \begin{align*}b\end{align*} is the base and \begin{align*}x\end{align*} is the exponent.
Logarithmic Form
\begin{align*}\log_b a=x\end{align*}, such that \begin{align*}b\end{align*} is the base.
Natural Log
The inverse of the natural number, \begin{align*}e\end{align*}, written \begin{align*}\ln\end{align*}.
Change of Base Formula
Let \begin{align*}b, x,\end{align*} and \begin{align*}y\end{align*} be positive numbers, \begin{align*}b \ne 1\end{align*} and \begin{align*}y \ne 1\end{align*}. Then, \begin{align*}\log_y x=\frac{\log_b x}{\log_b y}\end{align*}. More specifically, \begin{align*}\log_y x=\frac{\log x}{\log y}\end{align*} and \begin{align*}\log_y x=\frac{\ln x}{\ln y}\end{align*}, so that expressions can be evaluated using a calculator.

### Practice

Convert the following exponential equations to logarithmic equations.

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

Convert the following logarithmic equations to exponential equations.

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

convert the following logarithmic expressions without a calculator.

1. \begin{align*}\log_5 25\end{align*}
2. \begin{align*}\log_{\frac{1}{3}} 27\end{align*}
3. \begin{align*}\log \frac{1}{10}\end{align*}
4. \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.

1. \begin{align*}\log 72\end{align*}
2. \begin{align*}\ln 8\end{align*}
3. \begin{align*}\log_2 12\end{align*}
4. \begin{align*}\log_3 9\end{align*}
5. \begin{align*}\log_{11} 32\end{align*}

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Color Highlighted Text Notes

### Vocabulary Language: English

Change of Base Formula

Let $b, x,$ and $y$ be positive numbers, $b \ne 1$ and $y \ne 1$. Then, $\log_y x=\frac{\log_b x}{\log_b y}$. More specifically, $\log_y x=\frac{\log x}{\log y}$ and $\log_y x=\frac{\ln x}{\ln y}$, so that expressions can be evaluated using a calculator.

Exponential Form

The exponential form of an expression is $b^x=a$, where $b$ is the base and $x$ is the exponent.

Logarithm

A logarithm is the inverse of an exponential function and is written $\log_b a=x$ such that $b^x=a$.

Logarithmic Form

Logarithmic form is $\log_b a=x$, such that $b$ is the base.

Natural Log

A natural logarithm is a log with base $e$. The natural logarithm is written as ln.

Natural Logarithm

A natural logarithm is a log with base $e$. The natural logarithm is written as ln.

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