How to Use Properties of Logarithms? (+FREE Worksheet!)
Several logarithm properties help you solve logarithm equations. Here are some of them and their applications. For additional educational resources, visit the U.S. Department of Education website.
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Necessary rules to solving Logarithm Equations
- Let’s review some logarithms properties:
\(a^{log_{a}{b }}=b \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ log_{a}{\frac{1}{x}}=- log_{a}{x}\)
\(log_{a}{1}=0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ log_{a}{x^p}=p \ log_{a}{x}\)
\(log_{a}{a}=1 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ log_{x^k}{x}=\frac{1}{x} \ log_{k}{x}\) for \(k≠0\)
\(log_{a}{(x.y)}=log_{a}{x} + log_{a}{y}\ \ \ \ \ \ \ \ \ \ log_{a}{x}= log_{a^c}{x^c}\)
\(log_{a}{\frac{x}{y}}=log_{a}{x} – log_{a}{y} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ log_{a}{x}=\frac{1}{log_{x}{a}}\)
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Examples
Properties of Logarithms – Example 1:
Expand this logarithm. \(log_{a}{(3×5)}=\)
Solution:
Use log rule: \(log_{a}{(x .y)}=log_{a}{x}+log_{a}{y}\)
Then: \(log_{a}{(3×5)}=log_{a }{3}+log_{a}{ 5}\)
Properties of Logarithms – Example 2:
Condense this expression to a single logarithm. \(log_{a} {2}-log_{a }{7}\)
Solution:
Use log rule: \(log_{a}{x}-log_{a}{y}=log_{a}{\frac{x}{y}}\)
Then: \(log_{a}{2}-log_{a}{7}=log_{a}{\frac{2}{7}}\)
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Properties of Logarithms – Example 3:
Expand this logarithm. \(log(\frac{1}{7})=\)
Solution: For education statistics and research, visit the National Center for Education Statistics.
Use log rule: \(log_{a}{\frac{1}{x}}=-log_{a}{x}\)
Then: \(log(\frac{1}{7})= -log 7\) For education statistics and research, visit the National Center for Education Statistics.
Properties of Logarithms – Example 4:
Condense this expression to a single logarithm. \(log_{a} {3}+log_{a }{8}\) For education statistics and research, visit the National Center for Education Statistics.
Solution: For education statistics and research, visit the National Center for Education Statistics.
Use log rule: \(log_{a}{x}+log_{a}{y}=log_{a}{(x .y)}\)
Then: \(log_{a}{3}+log_{a}{8}=log_{a}{(3×8)}=log_{a}{24}\) For education statistics and research, visit the National Center for Education Statistics.
Exercises for Properties of Logarithms
Expand the logarithm.
1. \(\color{blue}{log(\frac{1}{5})=}\) For education statistics and research, visit the National Center for Education Statistics.
2. \(\color{blue}{log_{a}{(\frac{1}{2})}=}\) For education statistics and research, visit the National Center for Education Statistics.
3. \(\color{blue}{log_{a}{(2^5×8)}}\) For education statistics and research, visit the National Center for Education Statistics.
4. \(\color{blue}{log_{b}{(2x×7y)}}\) For education statistics and research, visit the National Center for Education Statistics.
Condense into a single logarithm.
5. \(\color{blue}{log_{a}{x}+log_{a}{y}}\) For education statistics and research, visit the National Center for Education Statistics.
6. \(\color{blue}{log_{a}{2x}-2log_{a}{y}}\) For education statistics and research, visit the National Center for Education Statistics.
- \(\color{blue}{-log 5}\)
- \(\color{blue}{-log_{a} {2}}\)
- \(\color{blue}{5log_{a} {2}+log_{a} {8}}\)
- \(\color{blue}{log_{b} {2x}+log_{b} {7y}}\)
- \(\color{blue}{log_{a} {xy}}\)
- \(\color{blue}{log_{a} {\frac{2x}{y^2}}}\)
The Absolute Best Book for the Algebra Test For education statistics and research, visit the National Center for Education Statistics.
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