Properties of Logarithms | Rules, Laws & Examples

1. Product Rule of Logarithms

Rule: \(\log_b(MN) = \log_b M + \log_b N\)

This rule means the log of a product equals the sum of the logs.

Example: \(\log_{10}(100 \times 10) = \log_{10} 100 + \log_{10} 10 = 2 + 1 = 3\)

2. Quotient Rule of Logarithms

Rule: \(\log_b\left(\frac{M}{N}\right) = \log_b M – \log_b N\)

This rule means the log of a division equals the difference of the logs.

Example: \(\log_{10}\left(\frac{100}{10}\right) = \log_{10} 100 – \log_{10} 10 = 2 – 1 = 1\)

3. Power Rule of Logarithms

Rule: \(\log_b\left(M^k\right) = k \times \log_b M\)

This rule allows exponents to be brought in front of the logarithm.

Example: \(\log_{10}\left(10^3\right) = 3 \times \log_{10} 10 = 3 \times 1 = 3\)

4. Change of Base Rule

Rule: \(\log_b M = \frac{\log_k M}{\log_k b}\)

This rule is useful when the base is not easily calculable.

Example: \(\log_{2} 8 = \frac{\log_{10} 8}{\log_{10} 2} = \frac{0.903}{0.301} = 3\)

5. Log of 1 Rule

Rule: \(\log_b 1 = 0\)

Since any number raised to the power of 0 is 1, the log of 1 is always 0.

Example: \(\log_{5} 1 = 0\)

6. Log of the Base Rule

Rule: \(\log_b b = 1\)

The logarithm of a number to its own base is always 1.

Example: \(\log_{7} 7 = 1\)