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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 referred to the logarithmic property: \( \log_b (x/y) = \log_b x - \log_b y \). This property simplifies divisions inside a logarithm into subtractions outside the logarithm. For instance, \( \log_b (a/c) \) can be rewritten as \( \log_b a - \log_b c \).

Step by step solution

01

Introducing Logarithmic Properties

Start with the common logarithmic property: \( \log_b (x \cdot y) = \log_b x + \log_b y \) which shows how multiplication inside the logarithm becomes addition outside. Similarly, division property exists. This is what the student is referring to.
02

Equation Transformation

The division property of logarithms can be written as follows: \( \log_b (x/y) = \log_b x - \log_b y \). When a division operation \(x/y\) is present inside the logarithm, it can be rewritten as a subtraction operation, subtracting the logarithm of the divisor from the logarithm of the dividend.
03

Illustration with an Example

For instance, consider \( \log_b (a/c) \). Using the logarithm division property, you can rewrite this as \( \log_b a - \log_b c \), simplifying the original expression.

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