<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
Skip Navigation

8.5: Defining Logarithms

Difficulty Level: At Grade Created by: CK-12
Atoms Practice
Estimated16 minsto complete
Practice Logarithms
This indicates how strong in your memory this concept is
Estimated16 minsto complete
Estimated16 minsto complete
Practice Now
This indicates how strong in your memory this concept is
Turn In

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?


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


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*}


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*}.


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.


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*}

Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes
Please to create your own Highlights / Notes
Show More


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.


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.

Image Attributions

Show Hide Details
Difficulty Level:
At Grade

Concept Nodes:

Date Created:
Mar 12, 2013
Last Modified:
Sep 07, 2016
Files can only be attached to the latest version of Modality
Please wait...
Please wait...
Image Detail
Sizes: Medium | Original