Solve for \( x \) : \( 6^{4 x-4}=2^{5 x-10} \) \( x=\square \) You may enter the exact value or round to 4 decimal places.

To solve for \( x \) in the equation \( 6^{4x-4} = 2^{5x-10} \), we start by expressing both sides in terms of base 2. First, note that \( 6 = 2 \cdot 3 \), so: \[ 6^{4x-4} = (2 \cdot 3)^{4x-4} = 2^{4x-4} \cdot 3^{4x-4} \] Now we can rewrite the equation: \[ 2^{4x-4} \cdot 3^{4x-4} = 2^{5x-10} \] Dividing both sides by \( 2^{4x-4} \) gives us: \[ 3^{4x-4} = 2^{5x-10-4x+4} = 2^{x-6} \] Next, we take logarithms to solve for \( x \): \[ \log(3^{4x-4}) = \log(2^{x-6}) \] Using the logarithmic identity \( \log(a^b) = b\log(a) \), we rewrite this as: \[ (4x-4) \log 3 = (x-6) \log 2 \] Expanding both sides yields: \[ 4x \log 3 - 4 \log 3 = x \log 2 - 6 \log 2 \] Rearranging gives: \[ 4x \log 3 - x \log 2 = -6 \log 2 + 4 \log 3 \] Factoring \( x \) out from the left side: \[ x (4 \log 3 - \log 2) = 4 \log 3 - 6 \log 2 \] Now, solving for \( x \): \[ x = \frac{4 \log 3 - 6 \log 2}{4 \log 3 - \log 2} \] Using logarithmic approximations, where \( \log 2 \approx 0.3010 \) and \( \log 3 \approx 0.4771 \): \[ x = \frac{4 \cdot 0.4771 - 6 \cdot 0.3010}{4 \cdot 0.4771 - 0.3010} \] \[ x = \frac{1.9084 - 1.8060}{1.9084 - 0.3010} \approx \frac{0.1024}{1.6074} \approx 0.0637 \] Thus, rounding to four decimal places, we have: \[ x \approx 0.0637 \]