Problem 91 - 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 statement the student made is referring to the Quotient Rule of logarithms, which states that division inside a log with the same base can be rewritten as subtraction. In formula terms, it's represented as \(\log_b (M/N) = \log_b M - \log_b N\).

Step 1: Identify the Quotient Rule of Logarithms

The Quotient Rule of logarithms states that the division of two numbers within a logarithm (with the same base) can be represented as the subtraction of two logarithmic expressions. The general equation for this property is \(\log_b (M/N) = \log_b M - \log_b N\). Here, 'b' is the base of the logarithm, 'M' is the numerator and 'N' is a denominator.

Step 2: Apply This Rule to a Specific Example

Suppose we have \(\log_2 (8/2)\). Applying the quotient rule, we can rewrite it as \(\log_2 8 - \log_2 2\). Solving this would give us \(3 - 1 = 2\), which is equal to \(\log_2 (8/2)\). This demonstrates that the division operation inside a logarithm has been successfully converted to a subtraction operation.