# Logarithms Properties

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

## 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.

1. $$log ⁡(8×5)=$$
2. $$log ⁡(9×4)=$$
3. $$log ⁡(3×7)=$$
4. $$\log{\frac{3}{4}}$$
5. $$\log{\frac{5}{7}}$$
6. $$\log({\frac{2}{5}})^3$$
7. $$log ⁡(2×3^4)=$$
8. $$\log({\frac{5}{7}})^4$$

1. $$log 8+log 5$$
2. $$log 9+log 9$$
3. $$log 3+log 7$$
4. $$log 3-log 4$$
5. $$log 5-log 7$$
6. $$3 log 2-3 log 5$$
7. $$log 2+4 log 3$$
8. $$4log 5-4 log 7$$

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