To use the Product Rule of Logarithms, recall that the Product Rule states:
\[ \log_b(MN) = \log_b(M) + \log_b(N) \]
For your specific example, you want to express \(\log_{24}(x)\) in terms of \(\log_{3}(x)\) and \(\log_{8}(x)\). Since \(24\) can be expressed as the product of \(3\) and \(8\) (i.e., \(24 = 3 \cdot 8\)), you can write:
\[ \log_{24}(24) = \log_{24}(3 \cdot 8) = \log_{24}(3) + \log_{24}(8) \]
However, since you are looking for an equivalent expression for \(\log_{24}\) in the options provided, you are trying to express it in terms of the products or sums of other logarithms rather than the direct application of the formula.
Thus, the correct expression that uses the Product Rule logically would be:
log 3 + log 8
So the answer is:
log 3 + log 8 (logarithm 3 plus logarithm 8).
1 answer