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:

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

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. Divide the Numbers:
   The quotient ( \frac{M}{N} ) can be written using exponents:

   $$\frac{M}{N} = \frac{b^x}{b^y} = b^{x-y}$$

3. Apply the Logarithm:
   Taking the logarithm of both sides:

   $$\log_b \left(\frac{M}{N}\right) = \log_b \left(b^{x-y}\right)$$

   Since the logarithm of (b^{x-y}) is simply (x-y), we have:

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

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