Solve, Round your answer to the nearest thousandth. e^x-4=9 x= square

To solve the equation $e^{x-4}=9$, we can take the natural logarithm of both sides to get rid of the exponential term.<br /><br />Taking the natural logarithm of both sides, we have:<br /><br />$\ln(e^{x-4}) = \ln(9)$<br /><br />Using the property of logarithms that $\ln(e^y) = y$, we can simplify the left side of the equation:<br /><br />$x-4 = \ln(9)$<br /><br />Now, we can solve for $x$ by adding 4 to both sides:<br /><br />$x = \ln(9) + 4$<br /><br />Using a calculator, we can find that $\ln(9) \approx 2.1972$. Therefore, the solution to the equation is:<br /><br />$x \approx 2.1972 + 4 = 6.1972$<br /><br />Rounding to the nearest thousandth, we have:<br /><br />$x \approx 6.197$<br /><br />Therefore, the answer is $x \approx 6.197$.