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# Logarithms

## Evaluate and convert logarithms to exponential form

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Defining Logarithms

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?

### Logarithm

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. We will now 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*}.

Let's rewrite log327=3\begin{align*}\log_3 27=3\end{align*} in exponential form.

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*}

Now, let's find the following.

1. log1000\begin{align*}\log 1000\end{align*}

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

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

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

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

log12(8)=x(12)x=8\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.

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

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

The first is logb1=0\begin{align*}\log_b 1=0\end{align*}, because b0=1\begin{align*}b^0=1\end{align*}. The second is logbb=1\begin{align*}\log_b b=1\end{align*} because b1=bb\begin{align*}b^1=b \cdot b\end{align*} can be any number except 1.

Finally, let's use a calculator to find the following logarithms and round our answers to the nearest hundredth.

1. ln7\begin{align*}\ln7\end{align*}

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.

1. log35\begin{align*}\log 35\end{align*}

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

1. log5226\begin{align*}\log_5 226\end{align*}

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: logax=logbxlogba\begin{align*}\log_a x=\frac{\log_b x}{\log_b a}\end{align*}, such that x,a,\begin{align*}x, a,\end{align*} and b>0\begin{align*}b>0\end{align*} and a\begin{align*}a\end{align*} and b1\begin{align*}b \ne 1\end{align*}.

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

log5226=log226log5\begin{align*}\log_5 226=\frac{\log 226}{\log 5}\end{align*} or ln226ln53.37\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.

### Examples

#### Example 1

Earlier, you were asked to find the decibel level of a concert if the intensity is 1,000,000,000(I0).

Plug the given values into the equation d=10logII0\begin{align*}d = 10 \cdot \log \frac{I}{I0}\end{align*} and solve for d.

d=10log1,000,000,000(I0)I0d=10log1,000,000,000d=109=90\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.

#### Example 2

Write 62=36\begin{align*}6^2=36\end{align*} in logarithmic form.

Using the key, we have: 62=36log636=2\begin{align*}6^2=36 \rightarrow \log_6 36=2\end{align*}.

For Examples 3-5, evaluate the expressions without a calculator.

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

#### Example 3

log1216\begin{align*}\log_{\frac{1}{2}} 16\end{align*}

log1216(12)x=16\begin{align*}\log_{\frac{1}{2}} 16 \rightarrow \left(\frac{1}{2}\right)^x=16\end{align*}. x\begin{align*}x\end{align*} must be negative because the answer is not a fraction, like the base.

24=16\begin{align*}2^4=16\end{align*}, so (12)4=16\begin{align*}\left(\frac{1}{2}\right)^{-4}=16\end{align*}. Therefore, log1216=4\begin{align*}\log_{\frac{1}{2}}16=-4\end{align*}.

#### log100\begin{align*}\log 100\end{align*}

log10010x=100\begin{align*}\log100 \rightarrow 10^x=100\end{align*}. \begin{align*}x=2\end{align*}, therefore, \begin{align*}\log100=2\end{align*}.

#### Example 5

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

\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*}.

#### Example 6

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

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*}.

### Review

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*}

To see the Review answers, open this PDF file and look for section 8.5.

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### 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.