How to Use Properties of Logarithms? (+FREE Worksheet!)

Several logarithms properties help you solve logarithm equations. Here are some of them and their applications.

How to Use Properties of Logarithms? (+FREE Worksheet!)

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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 \(k≠0\)
\(log_{a}{⁡(x.y)}=log_{a}{x} + log_{a}{y}\ \ \ \ \ \ \ \ \ \ log_{a⁡}{x}= log_{a^c}⁡{x^c}\)
\(log_{a}{⁡\frac{x}{y}}=log_{a}{x} – log_{a}{y} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ log_{a⁡}{x}=\frac{1}{log_{x}⁡{a}}\)

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: \(log_{a}{2}-log_{a}{7}=log_{a}{\frac{2}{7}}\)

Properties of Logarithms – Example 3:

Expand this logarithm. \(log(\frac{1}{7})=\)

Solution:

Use log rule: \(log_{a}{\frac{1}{x}}=-log_{a}{x}\)
Then: \(log(\frac{1}{7})= -log 7\)

Properties of Logarithms – Example 4:

Condense this expression to a single logarithm. \(log_{a} {3}+log_{a }{8}\)

Solution:

Use log rule: \(log_{a}{x}+log_{a}{y}=log_{a⁡}{(x .y)}\)
Then: \(log_{a}{3}+log_{a}{8}=log_{a}{⁡(3×8)}=log_{a}{⁡24}\)

Exercises for Properties of Logarithms

Expand the logarithm.

1. \(\color{blue}{log(\frac{1}{5})=}\)

2. \(\color{blue}{log_{a}{(\frac{1}{2})}=}\)

3. \(\color{blue}{log_{a}{(2^5×8)}}\)

4. \(\color{blue}{log_{b}{(2x×7y)}}\)

Condense into a single logarithm.

5. \(\color{blue}{log_{a}{x}+log_{a}{y}}\)

6. \(\color{blue}{log_{a}{2x}-2log_{a}{y}}\)

This image has an empty alt attribute; its file name is answers.png
  1. \(\color{blue}{-log 5}\)
  2. \(\color{blue}{-log_{a} {2}}\)
  3. \(\color{blue}{5log_{a} {2}+log_{a} {8}}\)
  4. \(\color{blue}{log_{b} {2x}+log_{b} {7y}}\)
  5. \(\color{blue}{log_{a} {xy}}\)
  6. \(\color{blue}{log_{a} {\frac{2x}{y^2}}}\)

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