What are the Laws of Logarithms?
The laws of logarithms are rules that help us simplify, expand, or contract logarithmic expressions. They follow directly from the laws of indices.
The Three Main Laws:
| Law | Formula | In Words |
|---|---|---|
| **Quotient Law** | \(\log_a \left(\frac{m}{n}\right) = \log_a m - \log_a n\) | Log of a quotient = difference of logs |
| **Power Law** | \(\log_a (m^n) = n \log_a m\) | Log of a power = power × log |
Why Do These Laws Work?
They are derived from the laws of indices:
- \(a^p \times a^q = a^{p+q}\) → adding logs corresponds to multiplication
- \(a^p \div a^q = a^{p-q}\) → subtracting logs corresponds to division
- \((a^p)^n = a^{pn}\) → multiplying log corresponds to power
Additional Important Rules:
| Rule | Formula | Example |
|---|---|---|
| Log of base | \(\log_a a = 1\) | \(\log_{10} 10 = 1\) |
| Change of base | \(\log_a b = \frac{\log_c b}{\log_c a}\) | \(\log_2 8 = \frac{\log 8}{\log 2}\) |