Quotient Rule

The logarithmic quotient rule states that

\[ log_x(\frac{a}{b}) = log_x(a) - log_x(b) \]

You can decompose the division of two inputs to a log function by subtracting the denominator from the numerator. The log functions must share the same base for this to work.

This works for the same reason that the product rule works. These are just an expression of exponents and when you divide one number with an exponent by another the exponent substract.

\[ 2^4 - 2^2 = 2^2 \]

\[ log_2(\frac{16}{4}) = log_2(16) - log_2(4) = 4 - 2 = 2 \]

Time Interpretation

You can interpret the division of log inputs as growing for a set of amount of time and then shrinking back in to another amount. The example below is the same as growing the amount for a period until it gets to \( 2 \) times its original size and then shrinking until the amount \( 3 \) times smaller than that. This makes sense because fractional exponents give you amounts in the past.