Logarithm Quotient Rule
The logarithm of a quotient equals the difference of logarithms.
Formula
Variables
Example
log₂(8/2) = log₂ 8 − log₂ 2 = 3 − 1 = 2.
How it works
This rule lets you turn a log of a division into the difference of two simpler logs, which is often easier to evaluate or simplify. It works because logarithms convert multiplication into addition, so division becomes subtraction: if x/y = x\cdot y^{-1}, then taking logs gives \log_b(x) + \log_b(y^{-1}), and the negative exponent becomes a subtraction. The base must be the same on both logs, and both x and y must be positive so the logarithms are defined.
When to use it
Use this when a logarithm contains a fraction and you want to simplify, expand, or calculate it step by step. It is also common in algebra, calculus, and science formulas when comparing ratios, such as in pH, decibels, or exponential growth models.
Worked example, step by step
Evaluate \(\log_5\left(\frac{125}{25}\right)\). \(\log_5\left(\frac{125}{25}\right) = \log_5(125) - \log_5(25)\) \(= 3 - 2\) \(= 1\) So, \(\log_5\left(\frac{125}{25}\right) = 1\).
Common mistakes to avoid
- Subtracting the numbers inside the logs instead of subtracting the log values
- Using the rule with different bases on each side
- Forgetting that x and y must both be positive
Did You Know?
Before calculators, slide rules used this rule physically — sliding two logarithmic scales to divide numbers.
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