Convert \( 100(1.36)^{x}=1850 \) to logarithmic form and use the change of base formula to solve for \( x \). Round answer to 3 decimal places. Hint: The change of base formula is \( \log _{b} a=\frac{\log a}{\log b} \). \( x= \)

To convert the equation \( 100(1.36)^{x}=1850 \) to logarithmic form, start by isolating \( (1.36)^{x} \): \[ (1.36)^{x} = \frac{1850}{100} = 18.5 \] Now, take the logarithm of both sides: \[ \log((1.36)^{x}) = \log(18.5) \] Using the power rule of logarithms, we can bring down the exponent: \[ x \cdot \log(1.36) = \log(18.5) \] Now, solve for \( x \): \[ x = \frac{\log(18.5)}{\log(1.36)} \] Now, apply the change of base formula. Assuming you're using base 10 for the logarithm: \[ \log(18.5) \approx 1.267 \] \[ \log(1.36) \approx 0.1335 \] Now, plug these values into the formula: \[ x = \frac{1.267}{0.1335} \approx 9.485 \] Rounding to three decimal places, we get: \[ \boxed{9.485} \]