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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 \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 @$\begin{align*}x=3\end{align*}@$ in @$\begin{align*}2^x=8\end{align*}@$ and @$\begin{align*}x=4\end{align*}@$ in @$\begin{align*}2^x=16\end{align*}@$ . But, what is @$\begin{align*}x\end{align*}@$ if @$\begin{align*}2^x=12\end{align*}@$ ? 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 @$\begin{align*}\log_b a=x\end{align*}@$ such that @$\begin{align*}b^x=a\end{align*}@$ . Therefore, if @$\begin{align*}5^2=25\end{align*}@$ ( exponential form ), then @$\begin{align*}\log_5 25=2\end{align*}@$ ( logarithmic form ).

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

Example A

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

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

@$$\begin{align*}\log_b a &= x \leftrightarrow b^x=a \\ \log_3 27 &= 3 \leftrightarrow 3^3=27 \end{align*}@$$

Example B

Find:

a) @$\begin{align*}\log 1000\end{align*}@$

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

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

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

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

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

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

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

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.

Answers

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 @$\begin{align*}\frac{\log \left(\frac{7}{9}\right)}{\log 8} \approx -0.12\end{align*}@$ .

Explore More

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. @$\begin{align*}\log_{11} 32\end{align*}@$

Vocabulary

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.

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