Problem 87 - Precalculus with Limits

Question

Rewrite each verbal statement as an equation. Then decide whether the statement is true or false. Justify your answer. The logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers.

Step-by-Step Solution

Answer

The statement is true. The equation representing the statement is \( \log_b(MN) = \log_b(M) + \log_b(N) \). The justification is based on the rules of exponents and commonly accepted laws of logarithms.

Step 1: Express the Statement as an Equation

The verbal statement can be translated into the mathematical form: \( \log_b(MN) = \log_b(M) + \log_b(N) \) for any base \(b\), and any positive numbers \( M, N \) which are not equal to 1. This rule is valid for any base of logarithm.

Step 2: Evaluate the Truth of the Statement

The statement is actually a classical property of logarithms. Therefore, it is indeed a true statement. It's the fundamental log rule, it is proved mathematically valid and widely used in algebra and calculus.

Step 3: Justification

This property of logarithms can be justified using the rules of exponents. If \( b^x = M \) and \( b^y = N \), then \( b^{x+y} = MN \). Taking log base \( b \) on both sides, we get \( x + y = \log_b(MN) \) which is equal to \( \log_b(M) + \log_b(N) \). Thus, the given statement is justified and proven to be true.