How to Use Properties of Logarithms

How to Use Properties of Logarithms

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

Related Topics

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  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  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  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. \(\color{blue}{log_{a} {\frac{2x}{y^2}}}\)

Related to "How to Use Properties of Logarithms"

Logarithms Properties
Logarithms Properties

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