Problem 140 - College Algebra Essentials

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

Yes, the statement makes sense. It aligns with the properties of logarithms wherein the bases and arguments have to be positive real numbers.

Step 1: Understanding Logarithms

Before starting, it's essential to recall the properties of a logarithm. The logarithm function is defined only for positive real numbers. This means, the base and the number of which we are taking logarithm (also referred as the 'argument' of the logarithm) must be positive.

Step 2: Evaluating the Statement

Now, looking at the given statement, 'It's important for check that the proposed solution of an equation with logarithms gives only logarithms of positive numbers in the original equation', we need to verify its logic by examining if it follows the properties of logarithms. As discussed, Logarithms should only be calculated with positive numbers, and any solution involving a logarithm should reflect this. This means a ‘proposed solution’ as mentioned in the statement, indeed needs to be evaluated to prevent any negative inputs for logarithms in the original equation.

Step 3: Making the Final Decision

After understanding the properties of logarithms and evaluating the statement with the properties, it's now evident that the statement makes sense. The statement correctly emphasizes the requirement of positive numbers for a logarithmic function, in line with the rules set out for logarithms.