Question 20 Find the solution of the exponential equation \( 9 e^{x}=15 \) in terms of logarithms, or correct to four decimal places. \( x=\square \)

To solve the equation \( 9 e^{x} = 15 \), start by isolating \( e^{x} \): \[ e^{x} = \frac{15}{9} = \frac{5}{3}. \] Now, take the natural logarithm of both sides: \[ x = \ln\left(\frac{5}{3}\right). \] Calculating this gives approximately: \[ x \approx \ln(1.6667) \approx 0.5110 \text{ (to four decimal places)}. \] For those looking to deepen their understanding of exponential equations, remember that these types of equations often model real-world phenomena, such as population growth or radioactive decay. Each distinct application might require a slight tweak in the approach, considering parameters like growth rates and decay constants! If you're tackling exponential equations regularly, a common pitfall is neglecting to isolate the exponential term before using logarithms. Taking it step by step can help avoid confusion, so always pull expressions apart carefully before diving into logs!