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

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

- \(\color{blue}{-log 5}\)
- \(\color{blue}{-log_{a} {2}}\)
- \(\color{blue}{5log_{a} {2}+log_{a} {8}}\)
- \(\color{blue}{log_{b} {2x}+log_{b} {7y}}\)
- \(\color{blue}{log_{a} {xy}}\)
- \(\color{blue}{log_{a} {\frac{2x}{y^2}}}\)

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