Determine the value of $\log_{3\sqrt{2}} \left( \frac{1}{18} \right)$.

Concept & Formula To evaluate a logarithm with complex bases and fractional arguments, set the expression to a variable, convert to exponential form, and isolate the power. $$ \log_a b = x \implies a^x = b $$ Step-by-Step Solution * **Given:** Evaluate $\log_{3\sqrt{2}} \left( \frac{1}{18} \right)$. * **Calculation:** Let the value be $x$. Convert to an exponential equation: $$ (3\sqrt{2})^x = \frac{1}{18} $$ * Square the base $(3\sqrt{2})$ to see its relationship with $18$: $$ (3\sqrt{2})^2 = 3^2 \cdot (\sqrt{2})^2 = 9 \cdot 2 = 18 $$ * Rewrite the right side of the equation ($\frac{1}{18}$) using negative exponents: $$ \frac{1}{18} = 18^{-1} $$ * Substitute $18$ with $(3\sqrt{2})^2$: $$ 18^{-1} = ((3\sqrt{2})^2)^{-1} = (3\sqrt{2})^{-2} $$ * Now, bring it back to our original equation: $$ (3\sqrt{2})^x = (3\sqrt{2})^{-2} $$ * Equate the exponents: $$ x = -2 $$ Exam Strategy & Shortcut Observe the relationship between $3\sqrt{2}$ and $18$. Squaring $3$ gives $9$, and squaring $\sqrt{2}$ gives $2$. $9 \times 2 = 18$. Because the argument is $1/18$, the exponent must be negative. Answer is $-2$. Common Pitfall Getting intimidated by the $3\sqrt{2}$ base. Don't panic—square it first to see if it relates to the argument. Most aptitude questions are designed with perfect squares or cubes in mind. Final Answer **Therefore, the correct answer is -2.**