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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 $d = 10 \cdot \log \frac{I}{I0}$ 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 $x=3$ in $2^x=8$ and $x=4$ in $2^x=16$ . But, what is $x$ if $2^x=12$ ? 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 $\log_b a=x$ such that $b^x=a$ . Therefore, if $5^2=25$ ( exponential form ), then $\log_5 25=2$ ( logarithmic form ).

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

#### Example A

Rewrite $\log_3 27=3$ in exponential form.

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

$\log_b a &= x \leftrightarrow b^x=a \\\log_3 27 &= 3 \leftrightarrow 3^3=27$

#### Example B

Find:

a) $\log 1000$

b) $\log_7 \frac{1}{49}$

c) $\log_{\frac{1}{2}}(-8)$

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

a) $\log 1000=x \Rightarrow 10^x=1000, x=3$ .

b) $\log_7 \frac{1}{49}=x \Rightarrow 7^x=\frac{1}{49}, x=-2$ .

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

#### Example C

a) $\ln7$

b) $\log 35$

c) $\log_5 226$

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

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

$\log_5 226=\frac{\log 226}{\log 5}$ or $\frac{\ln 226}{\ln 5} \approx 3.37$

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 $d = 10 \cdot \log \frac{I}{I0}$ and solve for d .

$d = 10 \cdot \log \frac{1,000,000,000 (I0)}{I0}\\d = 10 \cdot \log 1,000,000,000\\d = 10 \cdot 9 = 90$

Therefore, the decibel level of the concert is 90.

### Guided Practice

1. Write $6^2=36$ in logarithmic form.

2. Evaluate the following expressions without a calculator.

a) $\log_{\frac{1}{2}} 16$

b) $\log 100$

c) $\log_{64} \frac{1}{8}$

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

a) $\ln 32$

b) $\log_7 94$

c) $\log 65$

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

1. Using the key, we have: $6^2=36 \rightarrow \log_6 36=2$ .

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

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

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

b) $\log100 \rightarrow 10^x=100$ . $x=2$ , therefore, $\log100=2$ .

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

3. Using a calculator, we have:

a) 3.47 b) 2.33 c) 1.81

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

### Vocabulary

Logarithm
The inverse of an exponential function and is written $\log_b a=x$ such that $b^x=a$ .
Exponential Form
$b^x=a$ , such that $b$ is the base and $x$ is the exponent.
Logarithmic Form
$\log_b a=x$ , such that $b$ is the base.
Natural Log
The inverse of the natural number, $e$ , written $\ln$ .
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.

### Practice

Convert the following exponential equations to logarithmic equations.

1. $3^x=5$
2. $a^x=b$
3. $4(5^x)=10$

Convert the following logarithmic equations to exponential equations.

1. $\log_2 32=x$
2. $\log_{\frac{1}{3}}x=-2$
3. $\log_a y=b$

convert the following logarithmic expressions without a calculator.

1. $\log_5 25$
2. $\log_{\frac{1}{3}} 27$
3. $\log \frac{1}{10}$
4. $\log_2 64$

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

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

### Vocabulary Language: English

Change of Base Formula

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

Exponential Form

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

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

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

Natural Log

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

Natural Logarithm

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