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Chapter 12: Problem 96

Describe the quotient rule for logarithms and give an example.

Short Answer

Expert verified
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 by step solution

01

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.
02

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.
03

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, \( \log_2 (16/4) = 2 \).

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