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Logarithms

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Defining Logarithms
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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 = 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

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

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.

Answers

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

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