Problem 140 - College Algebra

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 because it underscores the importance of verifying solutions to logarithmic equations since the domain of the logarithmic function comprises only positive real numbers. Extraneous solutions can frequently occur leading to undefined or illegal operations if not thoroughly checked.

Step 1: Understanding the Logarithmic Function

Firstly, it's crucial to understand that the logarithm, as a function, is only defined for positive numbers. This is because the logarithmic functions 'log_b(x)' represents the power to which a fixed number (the base 'b') must be raised to obtain the number 'x'. Since we can't raise a number to any power to get a negative number or zero, 'x' can't be anything other than a positive number.

Step 2: Consideration in Solving Logarithmic Equations

When solving logarithmic equations, it is indeed very important to check that the solutions give only logarithms of positive numbers when substituted back into the original equation. If a given solution leads to a logarithm of a non-positive number, that solution is extraneous (not valid) and must be rejected.

Step 3: Validity of the Statement

Thus, the statement makes sense because it caution to the fact that while solving logarithmic equations, it’s crucial to check for potential extraneous solutions. It is a vital step to tag along because logarithmic equations often involve instances where an assumed solution might generate an undefined or illegal operation.