If $\log_3 x = -2$, then $x$ is equal to

Concept & Formula To find an unknown argument in a logarithmic equation, convert the logarithm into its exponential form. $$ \log_a b = x \implies a^x = b $$ A negative exponent indicates the reciprocal of the base raised to the positive exponent. $$ a^{-n} = \frac{1}{a^n} $$ Step-by-Step Solution * **Given:** The equation is $\log_3 x = -2$. * **Calculation:** Convert this logarithmic equation to an exponential equation using the base $3$: $$ 3^{-2} = x $$ * Apply the negative exponent rule to rewrite $3^{-2}$: $$ x = \frac{1}{3^2} $$ * Evaluate $3^2$: $$ x = \frac{1}{9} $$ Exam Strategy & Shortcut Whenever you see a negative result for a logarithm, it immediately means the argument is a fraction (less than $1$) if the base is greater than $1$. You know $3^2 = 9$, so the negative sign just flips it to $1/9$. You can eliminate options (a), (b), and (c) instantly because the argument of a logarithm (x) must always be positive. Common Pitfall A very common mistake is multiplying the base by the exponent to get $-6$, or raising it normally and then adding a negative sign to get $-9$. Always remember that a negative exponent creates a fraction, not a negative number. Final Answer **Therefore, the correct answer is \frac{1}{9}.**