Properties of Logarithms

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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: $$log 4-log 3= log{\frac{4}{3}}$$

Expand each logarithm.

1. $$\color{blue}{log ⁡(12×6)=}$$
2. $$\color{blue}{log ⁡(9×4)=}$$
3. $$\color{blue}{log ⁡(3×7)=}$$
4. $$\color{blue}{log{\frac{3}{4}}}$$
5. $$\color{blue}{log{\frac{5}{7}}}$$
6. $$\color{blue}{log({\frac{2}{5}})^3}$$
7. $$\color{blue}{log ⁡(2×3^4)=}$$
8. $$\color{blue}{ log({\frac{5}{7}})^4}$$

1. $$\color{blue}{log 12+log 6}$$
2. $$\color{blue}{ log 9+log 4}$$
3. $$\color{blue}{log 3+log 7}$$
4. $$\color{blue}{ log 3-log 4}$$
5. $$\color{blue}{ log 5-log 7}$$
6. $$\color{blue}{3 log 2-3 log 5}$$
7. $$\color{blue}{log 2+4 log 3}$$
8. $$\color{blue}{4log 5-4 log 7}$$

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