Problem 111 - College Algebra

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 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 1: 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.

Step 2: 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.

Step 3: 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.