Problem 101 - Introductory and Intermediate Algebra for College Students 4th

Question

You overhear a student talking about a property of logarithms in which division becomes subtraction. Explain what the student means by this.

Step-by-Step Solution

Answer

The property of logarithms where division becomes subtraction is known as the logarithm of a quotient rule. Mathematically, it's expressed as \( \log_a \frac{m}{n} = \log_a m - \log_a n \). It means that the logarithm of a quotient (division) is equal to the difference (subtraction) of the logarithms.

Step 1: Understand the Logarithm of a Quotient Rule

This property is also known as the logarithm of a quotient rule. The rule states that the logarithm of a quotient is equal to the difference of the logarithms. In mathematical terms, if a and b are positive numbers then the rule can be written as \( \log_a \frac{m}{n} = \log_a m - \log_a n \)

Step 2: Example

To illustrate what it means, suppose you have \( \log \frac{100}{10} \). According to the logarithm of a quotient rule, this can be written as \( \log 100 - \log 10 \). So, this property essentially transforms the division operation into a subtraction operation in the context of logarithms.