🔍 What is a Logarithm?

A logarithm answers the question: To what power must we raise a number (called the base) to get another number?

Mathematically:
$ \log_b(x) = y \Rightarrow b^y = x $

Example:
$ \log_2(8) = 3 \quad \text{because} \quad 2^3 = 8 $

1. Product Rule

$ \log_b(M \cdot N) = \log_b(M) + \log_b(N) $

Example:
$ \log_{10}(100 \cdot 1000) = \log_{10}(100) + \log_{10}(1000) = 2 + 3 = 5 $

2. Quotient Rule

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

Example:
$ \log_{10}\left(\frac{1000}{100}\right) = \log_{10}(1000) – \log_{10}(100) = 3 – 2 = 1 $

3. Power Rule

$ \log_b(M^n) = n \cdot \log_b(M) $

Example:
$ \log_2(8^2) = 2 \cdot \log_2(8) = 2 \cdot 3 = 6 $

4. Change of Base Rule

$ \log_b(x) = \frac{\log_a(x)}{\log_a(b)} $

Example:
$ \log_2(8) = \frac{\log_{10}(8)}{\log_{10}(2)} \approx \frac{0.903}{0.301} = 3 $

5. Special Values

$ \log_b(1) = 0 \quad \text{and} \quad \log_b(b) = 1 $

Examples:
$ \log_5(1) = 0 $
$ \log_7(7) = 1 $