What is the basis for the logarithmic equation logbA x C?

To understand why the correct answer is the sum of logarithms, it's essential to recall one of the fundamental properties of logarithms. Specifically, the property states that the logarithm of a product can be expressed as the sum of the logarithms of the individual factors. In mathematical terms, this is articulated as:

log_b(A * C) = log_bA + log_bC

When you have a logarithmic equation in the form of log_b(A * C), you're essentially indicating the logarithm of the product of A and C with respect to the base b. Applying the aforementioned property directly allows us to deconstruct the expression neatly into two parts, log_bA and log_bC.

Thus, when encountering log_b(A * C), you'll find that it equates to log_bA + log_bC, which confirms why this choice is the appropriate basis for the logarithmic equation provided. Understanding this logarithmic property is crucial in simplifying and manipulating logarithmic expressions effectively.