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Chapter 4: Problem 111

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

Short Answer

Expert verified
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 by step solution

01

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.
02

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.
03

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 \( 4 - 2=2 \).

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