Properties of Logarithms

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

Step by step guide to Properties of Logarithms

Learn some logarithms properties:

1. \(a^{\log_{a}{b}}=b\)
2. \(\log_{a}{1}=0\)
3. \(\log_{a}{a}=1\)
4. \(\log_{a}{x.y}=\log_{a}{x}+\log_{a}{y}\)
5. \(\log_{a}{\frac{x}{y}}=\log_{a}{x}-\log_{a}{y}\)
6. \(\log_{a}{\frac{1}{x}}=-\log_{a}{x}\)
7. \(\log_{a}{x^p}=p\log_{a}{x}\)
8. \(\log_{x^k}{x}=\frac{1}{x}\log_{a}{x}, for k\neq0\)
9. \(\log_{a}{x}=\log_{a^c}{x^c}\)
10. \(\log_{a}{x}=\frac{1}{\log_{x}{a}}\)

Example 1:

Expand this logarithm. \(log ⁡(8×5)=\)

Answer:

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

then: \(log ⁡(8×5)= log 8 + log 5\)

Example 2:

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

Answer:

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

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

Example 3:

Expand this logarithm. \(log ⁡(2×3)=\)

Answer:

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

then: \(log ⁡(2×3)= log 2 + log 3\)

Example 4:

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

Answer:

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

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

Exercises

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