Logarithm Quotient Rule

The Quotient Rule of logarithms states that the logarithm of a quotient is equal to the difference between the logarithm of the numerator and the logarithm of the denominator. In mathematical terms:

Why this works

1. Definition of Logarithms:
   If \(\log_b M = x\) and \(\log_b N = y\), then by definition, \(M = b^x\) and \(N = b^y\).

2. \[ \frac{M}{N} = \frac{b^x}{b^y} = b^{x-y} \]
3. \[ \log_b \left(\frac{M}{N}\right) = \log_b \left(b^{x-y}\right) \]\[ \log_b \left(\frac{M}{N}\right) = x - y = \log_b M - \log_b N \]

This rule is extremely useful for simplifying expressions involving logarithms and solving equations where logarithms appear.

Latex

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