Solve $7^{3 x-4}=3^{x-5}$ for $x$. Give both the exact answer and the decimal approximation to the nearest hundredth. If logarithms are needed to solve the equation, use "In."

Exact answer:

Decimal approximation:

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Answer

$\boxed{x \approx 3.47}$

Steps

Step 1 ：$\ln(7^{3x-4}) = \ln(3^{x-5})$

Step 2 ：$(3x-4)\ln(7) = (x-5)\ln(3)$

Step 3 ：$3x\ln(7) - 4\ln(7) = x\ln(3) - 5\ln(3)$

Step 4 ：$3x\ln(7) - x\ln(3) = 5\ln(3) + 4\ln(7)$

Step 5 ：$x(3\ln(7) - \ln(3)) = 5\ln(3) + 4\ln(7)$

Step 6 ：$x = \frac{5\ln(3) + 4\ln(7)}{3\ln(7) - \ln(3)}$

Step 7 ：$x \approx 3.47$

Step 8 ：$\boxed{x \approx 3.47}$