Problem 140 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. It's important for me to check that the proposed solution of an equation with logarithms gives only logarithms of positive numbers in the original equation.

Step-by-Step Solution

Answer

The statement makes sense as it is crucial to verify the solutions of a logarithmic equation to ensure logarithms are defined for those values, i.e., the values are positive.

Step 1: Understand the statement

The statement is about the solution process in logarithmic equations. Fundamentally, we know that logarithms are only defined for positive numbers.

Step 2: Analyze the statement

In this step, the statement's logical validity needs to be analyzed. It emphasizes the importance of checking the consequent values after solving logarithmic equations.

Step 3: Determine validity

Given the properties of logarithms, the statement makes sense. After finding the potential solutions of a logarithmic equation, they must be substituted back into the original equation to ensure validity because logarithms can only take positive numbers as input.