Solve the equation analytically. (Analytically means with algebraic methods.) Write the answer in exact form using fractions, not decimals. Your answer will also contain a log or natural log. \( 4 e^{5 x}=36 \) \( x=\square \) Write "log_b(a)" for \( \log _{b}(a) \) or " \( \ln (a) \) " for \( \ln (a) \).

To solve the equation \( 4 e^{5x} = 36 \), first, divide both sides by 4: \[ e^{5x} = \frac{36}{4} = 9. \] Next, take the natural logarithm of both sides: \[ \ln(e^{5x}) = \ln(9). \] Using the property of logarithms that states \( \ln(e^y) = y \): \[ 5x = \ln(9). \] Now, divide both sides by 5 to isolate \( x \): \[ x = \frac{\ln(9)}{5}. \] Thus, the solution to the equation is: \[ x = \frac{\ln(9)}{5}. \]