Solve for \( x \) \[ 15^{4 x}=4^{x-5} \] Round your answer to the nearest thousandth. Do not round any intermediate computations.

To solve the equation \( 15^{4x} = 4^{x-5} \), we can start by taking the logarithm of both sides. We'll use natural logarithm for this example, but you could use logarithm to any base. First, rewrite the equation: \[ \ln(15^{4x}) = \ln(4^{x-5}) \] Using the power rule of logarithms, we can simplify both sides: \[ 4x \ln(15) = (x-5) \ln(4) \] Now, distribute \( \ln(4) \) on the right side: \[ 4x \ln(15) = x \ln(4) - 5 \ln(4) \] Next, we rearrange the equation to get all terms involving \( x \) on one side: \[ 4x \ln(15) - x \ln(4) = - 5 \ln(4) \] Factor out \( x \) from the left side: \[ x (4 \ln(15) - \ln(4)) = -5 \ln(4) \] Now, solve for \( x \): \[ x = \frac{-5 \ln(4)}{4 \ln(15) - \ln(4)} \] Now we can calculate the value of \( x \). First, compute the logarithms: \[ \ln(4) \approx 1.386294361 \] \[ \ln(15) \approx 2.708050201 \] Now plug these values into the equation for \( x \): \[ 4 \ln(15) \approx 4 \times 2.708050201 = 10.832200804 \] Now find \( 4 \ln(15) - \ln(4) \): \[ 4 \ln(15) - \ln(4) \approx 10.832200804 - 1.386294361 \approx 9.445906443 \] Substituting back into our equation for \( x \): \[ x \approx \frac{-5 \times 1.386294361}{9.445906443} \approx \frac{-6.931471805}{9.445906443} \approx -0.7334796673 \] Rounding to the nearest thousandth, we find: \[ \boxed{-0.733} \]