Set 5: Exponential Functions (Intermediate)

Choice D is correct. Choice D is the correct answer. Isolate the exponential and solve. 1. Divide: $e^{2 x} = \frac{125}{5} = 25$. 2. Natural log: $\ln(e^{2 x}) = \ln(25)$. 3. Simplify: $2x = \ln(25) = \ln(5^2) = 2\ln(5)$. 4. Solve: $x = \frac{2\ln(5)}{2} = \ln(5)$. 5. Verify: $5e^{2\ln(5)} = 5 e^{\ln(25)} = 5(25) = 125$ ✓ Strategic Tip: Simplify logarithms using $\ln(a^n) = n\ln(a)$. Choice A is incorrect because $\ln(125) = \ln(5^3) = 3\ln(5)$, giving $x = \frac{3\ln(5)}{2}$. Choice B is incorrect because $x = \ln(25)$ gives $e^{2 x} = e^{2\ln(25)} = 625$. Choice C is incorrect because this uses arithmetic instead of logarithms.