# Properties of Logarithms

Do you want to know how to Properties of Logarithms? you can do it in two easy steps.

## Related Topics

- How to Solve Logarithmic Equations
- How to Solve Natural Logarithms Problems
- How to Evaluate Logarithms

## Properties of Logarithms

Learn some logarithms properties:

- \(a^{log_{a}{b}}=b\)
- \(log_{a}{1}=0\)
- \(log_{a}{a}=1\)
- \(log_{a}{x.y}=log_{a}{x}+\log_{a}{y}\)
- \(log_{a}{\frac{x}{y}}=log_{a}{x}-\log_{a}{y}\)
- \(log_{a}{\frac{1}{x}}=-log_{a}{x}\)
- \(log_{a}{x^p}=p log_{a}{x}\)
- \(log_{x^k}{x}=\frac{1}{x}log_{a}{x}\), for \(k\neq0\)
- \(log_{a}{x}= log_{a^c}{x^c}\)
- \(log_{a}{x}=\frac{1}{log_{x}{a}}\)

### Logarithms Properties – Example 1:

Expand this logarithm. \(log (8×5)=\)

**Solution:**

Use log rule: \(log_{a}{x.y}=log_{a}{x}+\log_{a}{y}\)

then: \(log (8×5)= log 8 + log 5\)

### Logarithms Properties – Example 2:

Condense this expression to a single logarithm. \(log 2-log 9=\)

**Solution:**

Use log rule: \(log_{a}{\frac{x}{y}}=log_{a}{x}-\log_{a}{y}\)

then: \(log 2-log 9= log{\frac{2}{9}}\)

### Logarithms Properties – Example 3:

Expand this logarithm. \(log (2×3)=\)

**Solution:**

Use log rule: \(log_{a}{x.y}=log_{a}{x}+\log_{a}{y}\)

then: \(log (2×3)= log 2 + log 3\)

### Logarithms Properties – Example 4:

Condense this expression to a single logarithm. \(log 4-log 3=\)

**Solution:**

Use log rule: \(log_{a}{\frac{x}{y}}=log_{a}{x}-log_{a}{y}\)

then: \(log 4-log 3= log{\frac{4}{3}}\)

## Exercises for Logarithms Properties

## Expand each logarithm.

- \(\color{blue}{log (12×6)=}\)
- \(\color{blue}{log (9×4)=}\)
- \(\color{blue}{log (3×7)=}\)
- \(\color{blue}{log{\frac{3}{4}}}\)
- \(\color{blue}{log{\frac{5}{7}}}\)
- \(\color{blue}{log({\frac{2}{5}})^3}\)
- \(\color{blue}{log (2×3^4)=}\)
- \(\color{blue}{ log({\frac{5}{7}})^4}\)

## Answers

- \(\color{blue}{log 12+log 6}\)
- \(\color{blue}{ log 9+log 4}\)
- \(\color{blue}{log 3+log 7}\)
- \(\color{blue}{ log 3-log 4}\)
- \(\color{blue}{ log 5-log 7}\)
- \(\color{blue}{3 log 2-3 log 5}\)
- \(\color{blue}{log 2+4 log 3}\)
- \(\color{blue}{4log 5-4 log 7}\)

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