$$ \log _{6}\left(3 x^{3}\right)=3 \log _{6}(3 x) $$ 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.

Question

UpStudy AI · Accepted Answer

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).
The statement is false. The correct statement is $$ \log_{6}(3) = 3 \log_{6}(3) $$ (which is not true).

\( \log _{6}\left(3 x^{3}\right)=3 \log _{6}(3 x) \) 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.

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