Solve the following exponential equation. Express your answer as a decimal approximation rounded to two decimal places.
\[
e^{x-2}=11^{\frac{2 x}{11}}
\]
Answer
Compute the right-hand side to get the final answer: \(\boxed{x \approx 1.85}\)
Step 1 ：Take the natural logarithm on both sides of the equation: \(\ln(e^{x-2}) = \ln(11^{\frac{2x}{11}})\)
Step 2 ：Use the property of logarithms to bring the exponent to the front: \((x-2) = \frac{2x}{11} \ln(11)\)
Step 3 ：Multiply through by 11 to clear the fraction: \(11x - 22 = 2x \ln(11)\)
Step 4 ：Rearrange terms: \(11x - 2x \ln(11) = 22\)
Step 5 ：Factor out x: \(x (11 - 2 \ln(11)) = 22\)
Step 6 ：Solve for x: \(x = \frac{22}{11 - 2 \ln(11)}\)
Step 7 ：Compute the right-hand side to get the final answer: \(\boxed{x \approx 1.85}\)
