1. **State the problem:** We want to simplify the expression $e^{\ln a^x}$. 2. **Recall the properties of logarithms and exponentials:** The natural logarithm $\ln$ and the exponential function $e^x$ are inverse functions. This means that for any positive number $y$, $e^{\ln y} = y$. 3. **Apply the logarithm power rule:** The expression inside the logarithm is $a^x$. Using the power rule for logarithms, we have: $$\ln a^x = x \ln a$$ 4. **Rewrite the original expression:** Substitute $\ln a^x$ with $x \ln a$: $$e^{\ln a^x} = e^{x \ln a}$$ 5. **Use the property of exponentials:** We can rewrite $e^{x \ln a}$ as: $$e^{x \ln a} = (e^{\ln a})^x$$ 6. **Simplify using the inverse property:** Since $e^{\ln a} = a$, we get: $$ (e^{\ln a})^x = a^x$$ **Final answer:** $$e^{\ln a^x} = a^x$$ This shows that $e^{\ln a^x}$ simplifies directly to $a^x$ by using the inverse properties of the exponential and logarithm functions.