Solve the Logarithm Equation: x∙log_m(1/3^x)

To solve this problem, we will apply the rules of logarithms as follows:

• Firstly, rewrite the expression $\log_m \frac{1}{3^x}$ using the Quotient Rule:

$\log_m \frac{1}{3^x} = \log_m 1 - \log_m 3^x$

• Since $\log_m 1 = 0$, the expression simplifies to:

$0 - \log_m 3^x = -\log_m 3^x$

• Apply the Power Rule to simplify $-\log_m 3^x$:

$-\log_m 3^x = -x \log_m 3$

• Substitute back to the original expression $x \log_m \frac{1}{3^x}$:

$x ( -x \log_m 3) = -x^2 \log_m 3$

Therefore, the solution to the problem in terms of simplifying the expression is $-x^2 \log_m 3$.