Problem 96 - Introductory and Intermediate Algebra for College Students 4th

Question

Describe the quotient rule for logarithms and give an example.

Step-by-Step Solution

Answer

The quotient rule for logarithms is a rule that states the logarithm of a quotient is equivalent to the difference of the logarithms of the numerator and the denominator. It is expressed as: \( \log_b (a/b) = \log_b(a) - \log_b(b) \).

Step 1: Quotient Rule for Logarithms

The quotient rule for logarithms is a rule in mathematics that explains how to simplify the logarithm of a quotient (division of two numbers). The rule states that the logarithm of a quotient is equivalent to the difference of the logarithms of the numerator and the denominator. Mathematically, this is expressed as: \( \log_b (a/b) = \log_b(a) - \log_b(b) \). 'a' and 'b' represent the numbers you're dividing, and 'b' is also the base of the logarithm.

Step 2: Explanation

In simple terms, the quotient rule tells us that the logarithm of a division can be rewritten as subtraction of logarithms. This rule is applied while simplifying logarithm expressions.

Step 3: Example of the Quotient Rule

Let's look at an example to better understand this rule: Consider the expression \( \log_2 (16/4) \). According to the quotient rule, this expression can be rewritten as: \( \log_2(16) - \log_2(4) \). Evaluating this, \(\log_2(16) = 4\) and \(\log_2(4) = 2\). So, performing the subtraction, this simplifies to \(4 - 2 = 2\). Thus, \( \log_2 (16/4) = 2 \).