\[ 3^{x}=\frac{1}{81} \] Determine which of the following equations has an equivalent solution to the equation given above. A. \( 5^{x}=\frac{1}{125} \) B. \( 6^{-x}=\frac{1}{1296} \) C. \( 7^{(x-1)}=7^{(2 x+3)} \) D. \( 2^{3 x}=2^{(5 x-8)} \)

To solve the equation \( 3^{x}=\frac{1}{81} \), we can rewrite \(\frac{1}{81}\) as \(3^{-4}\) (since \(81 = 3^4\)). So we have: \[ 3^{x} = 3^{-4} \] This means that \(x = -4\). Now let's check the other equations: A. For \( 5^{x} = \frac{1}{125} \), we can write \( \frac{1}{125} = 5^{-3} \). Thus, \(x = -3\), which is not equivalent. B. For \( 6^{-x} = \frac{1}{1296} \), we know \(1296 = 6^4\). So, \(\frac{1}{1296} = 6^{-4}\) implies \(-x = -4\), giving \(x = 4\), which is not equivalent. C. The equation \(7^{(x-1)}=7^{(2 x+3)}\) implies \(x-1 = 2x + 3\). Solving for \(x\): \[ x - 2x = 3 + 1 \implies -x = 4 \implies x = -4 \] This is equivalent! D. For \( 2^{3x} = 2^{(5x-8)} \), we have \(3x = 5x - 8\). Rearranging gives \(2x = 8\) and \(x = 4\), which is not equivalent. So, the equation with an equivalent solution is option C: \( 7^{(x-1)}=7^{(2 x+3)} \). In summary: - The solution provided gives you the equivalent equation for \(x = -4\). - While working with exponential equations, remember that equal bases allow you to set exponents equal to one another—a key strategy that can save you time on tests!