What is the formula for the logarithm of a quotient?

The formula for the logarithm of a quotient states that the logarithm of a division can be expressed in terms of the logarithms of the numerator and the denominator. Specifically, the correct formula is expressed as the logarithm of the first value minus the logarithm of the second value. This means that if you have two positive numbers (x) and (y), the logarithm of their quotient (x/y) can be calculated by taking the logarithm of (x) and subtracting the logarithm of (y).

This property arises from the definition of logarithms and their relationship to exponents. When (b^{log_b(x)}) is equal to (x) and (b^{log_b(y)}) is equal to (y), dividing these two expressions, we realize that (log_b(x/y)) captures the decrease in value due to the division of (x) by (y), thereby resulting in a subtraction of logarithms.

An understanding of this fundamental property is crucial for simplifying logarithmic expressions and solving logarithmic equations effectively. It plays an important role in many areas of mathematics and its applications.