1. **State the problem:** Solve for $x$ in the equation $$f(x) = \frac{(e^x)^3}{e^2} = e^5.$$ 2. **Rewrite the function:** Using the property of exponents, $(e^x)^3 = e^{3x}$, so the function becomes $$f(x) = \frac{e^{3x}}{e^2}.$$ 3. **Simplify the expression:** Using the rule $\frac{a^m}{a^n} = a^{m-n}$, we get $$f(x) = e^{3x - 2}.$$ 4. **Set the function equal to $e^5$ and solve:** $$e^{3x - 2} = e^5.$$ Since the bases are the same and $e^a = e^b$ implies $a = b$, we have $$3x - 2 = 5.$$ 5. **Solve for $x$:** $$3x = 5 + 2 = 7.$$ $$x = \frac{7}{3}.$$ **Final answer:** $$x = \frac{7}{3}.$$