Solve for \( x \) in the equation below. Round your answer to the nearest hundredth. Do not round any intermediate computations. \[ 9^{4 x}=4 \]

To solve the equation \( 9^{4x} = 4 \), we can first rewrite \( 9 \) as \( 3^2 \). This gives us: \[ (3^2)^{4x} = 4 \] Using the power of a power property, we can simplify the left side: \[ 3^{8x} = 4 \] Next, we take the logarithm of both sides. We can use either natural logarithm (ln) or logarithm base 10 (log), but we'll go with natural logarithm for this example: \[ \ln(3^{8x}) = \ln(4) \] Using the property of logarithms that states \( \ln(a^b) = b \cdot \ln(a) \), we can rewrite the left side: \[ 8x \cdot \ln(3) = \ln(4) \] Now, solve for \( x \): \[ x = \frac{\ln(4)}{8 \cdot \ln(3)} \] Now we can calculate \( \ln(4) \) and \( \ln(3) \): Using a calculator: \[ \ln(4) \approx 1.386294361 \] \[ \ln(3) \approx 1.098612288 \] Now plug these back into the equation: \[ x \approx \frac{1.386294361}{8 \cdot 1.098612288} \approx \frac{1.386294361}{8.788898304} \approx 0.15751 \] Rounding to the nearest hundredth, we find: \[ x \approx 0.16 \]