Logarithms Properties

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.

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

Answers

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. \(4log 5-4 log 7\)