Full solution

4^x=7^{x-4}

Identify Equation: Identify the equation to solve: $4^x = 7^{x-4}$ .
Take Natural Logarithm: Take the natural logarithm (ln) of both sides to bring down the exponents: $\ln(4^x) = \ln(7^{x-4})$ .
Apply Power Rule: Apply the power rule of logarithms: $x\ln(4) = (x-4)\ln(7)$ .
Distribute $\ln(7)$ : Distribute $\ln(7)$ on the right side: $x\cdot\ln(4) = x\cdot\ln(7) - 4\cdot\ln(7)$ .
Move Terms: Move $x$ terms to one side and constants to the other: $x\ln(4) - x\ln(7) = -4\ln(7)$ .
Factor Out x: Factor out $x$ from the left side: $x(\ln(4) - \ln(7)) = -4\ln(7)$ .
Divide Both Sides: Divide both sides by $(\ln(4) - \ln(7))$ to solve for $x$ : $x = \frac{-4\cdot\ln(7)}{\ln(4) - \ln(7)}$ .
Calculate Value of x: Calculate the value of $x$ using a calculator: $x \approx -4 \times 1.94591 / (1.38629 - 1.94591)$ .
Finish Calculation: Finish the calculation: $x \approx -4 \times 1.94591 / (-0.55962)$ .
Get Final Answer: Get the final answer: $x \approx -4 \times -3.4785$ .
Get Final Answer: Get the final answer: $x \approx -4*-3.4785$ .Complete the calculation: $x \approx 13.914$ .