91Ó°ÊÓ

To find the value of x, we can take the natural logarithm (base e) of both sides of the equation \(3^x = 4\). This gives us the equation \[x\ln{3} = \ln{4}\] Now we can solve for x by dividing both sides of the equation by \(\ln{3}\): \[x = \frac{\ln{4}}{\ln{3}}\]

Now that we have the value of x, we can use it to evaluate the expression \(3^{-2x}\). To do this, first replace x with the expression we found: \[3^{-2\left(\frac{\ln{4}}{\ln{3}}\right)}\] Now we can simplify the expression using the properties of exponents. The exponent is \(-2\left(\frac{\ln{4}}{\ln{3}}\right)\), so we can write: \[3^{-2\left(\frac{\ln{4}}{\ln{3}}\right)} = \left(3^{\frac{\ln{4}}{\ln{3}}}\right)^{-2}\] Since we know that \(3^{\frac{\ln{4}}{\ln{3}}} = 4\), we can substitute this value back into the expression: \[\left(4\right)^{-2}\] Now, we can find the final result by evaluating the expression: \[\left(4\right)^{-2} = \frac{1}{4^2} = \frac{1}{16}\] So, the value of \(3^{-2x}\) when \(3^x = 4\) is \(\frac{1}{16}\).