\( \log _{6}\left(3 x^{3}\right)=3 \log _{6}(3 x) \) Select the correct choice below and, if necessary, fill in the answer box to complet A. The statement is false. The correct statement is \( \square=3 \log _{6}(3 x) \) B. The statement is true.

To determine whether the statement \( \log _{6}\left(3 x^{3}\right)=3 \log _{6}(3 x) \) is true or false, we can simplify both sides of the equation. Starting with the left side: \[ \log_{6}(3x^{3}) = \log_{6}(3) + \log_{6}(x^{3}) = \log_{6}(3) + 3\log_{6}(x) \] Now, let's simplify the right side: \[ 3 \log_{6}(3x) = 3(\log_{6}(3) + \log_{6}(x)) = 3\log_{6}(3) + 3\log_{6}(x) \] Now we can compare both sides: Left side: \[ \log_{6}(3) + 3\log_{6}(x) \] Right side: \[ 3\log_{6}(3) + 3\log_{6}(x) \] For the two sides to be equal, we need: \[ \log_{6}(3) + 3\log_{6}(x) = 3\log_{6}(3) + 3\log_{6}(x) \] Subtracting \(3\log_{6}(x)\) from both sides gives: \[ \log_{6}(3) = 3\log_{6}(3) \] This simplifies to: \[ \log_{6}(3) - 3\log_{6}(3) = 0 \implies -2\log_{6}(3) = 0 \] This implies that \(\log_{6}(3) = 0\), which is not true since \(\log_{6}(3)\) is a positive number. Thus, the original statement is false. The correct choice is: **A. The statement is false. The correct statement is \( \log_{6}(3) = 3 \log_{6}(3) \)** (which is not true).