Problem 111 - College Algebra Essentials

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 student is referring to the Quotient Rule of logarithms. This rule states that the logarithm of a quotient is equal to the subtraction of the logarithm of the numerator and the logarithm of the denominator.

Step 1: Understanding the Quotient Rule

The Quotient Rule is a part of basic logarithm properties. It states that the logarithm of a quotient is equal to the logarithm of the numerator subtracted by the logarithm of the denominator. In other words, \( log_b(\frac{m}{n}) = log_b(m) - log_b(n) \) where b is the base of the logarithm, and m and n are any two positive real numbers.

Step 2: Proving the Quotient Rule

This rule can be derived from the basic properties of logarithms. Given \( m = b^{log_b(m)} \) and \( n = b^{log_b(n)} \), you can replace m and n in the quotient (m/n) with these expressions. You will get \(\frac{b^{log_b(m)}}{b^{log_b(n)}}\), which is equal to \( b^{log_b(m) - log_b(n)} \) due to the laws of exponents. But by the definition of logarithms, \( b^{log_b(m) - log_b(n)} \) is equal to \( \frac{m}{n} \). Hence, the logarithm of a quotient is the difference between the logarithms of m and n.

Step 3: Providing an example

Consider the following example to understand how the rule works: Let's say we want to calculate \( log_2(\frac{16}{4}) \). According to the Quotient Rule, this would be equal to \( log_2(16) - log_2(4) \) , which finally simplifies to \( 4 - 2=2 \).