Properties of Logarithms

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.

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. \(\color{blue}{ log({\frac{5}{7}})^4}\)

Answers

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. \(\color{blue}{4log 5-4 log 7}\)