Question 8 Solve the equation \( 4^{(x+5)}=7^{(x-2)} \). Leave your answer in terms of logarithms (do not provide a decimal answer).

We start with the equation \[ 4^{(x+5)} = 7^{(x-2)}. \] Taking the natural logarithm of both sides gives \[ \ln\left(4^{(x+5)}\right) = \ln\left(7^{(x-2)}\right). \] Using the logarithm power rule, which states that \(\ln(a^b) = b\ln(a)\), we can rewrite the equation as \[ (x+5)\ln(4) = (x-2)\ln(7). \] Expanding both sides yields \[ x\ln(4) + 5\ln(4) = x\ln(7) - 2\ln(7). \] To isolate the terms with \(x\), subtract \(x\ln(7)\) from both sides: \[ x\ln(4) - x\ln(7) + 5\ln(4) = -2\ln(7). \] Next, subtract \(5\ln(4)\) from both sides: \[ x\ln(4) - x\ln(7) = -2\ln(7) - 5\ln(4). \] Factor out \(x\) from the left-hand side: \[ x\left(\ln(4) - \ln(7)\right) = -2\ln(7) - 5\ln(4). \] Finally, solve for \(x\) by dividing both sides by \(\ln(4) - \ln(7)\): \[ x = \frac{-2\ln(7) - 5\ln(4)}{\ln(4) - \ln(7)}. \] This is the solution in terms of logarithms. Alternatively, by multiplying the numerator and denominator by \(-1\), we can also write the answer as \[ x = \frac{2\ln(7) + 5\ln(4)}{\ln(7) - \ln(4)}. \]