The expression \( e^{\ln{x}} \) simplifies to just \( x \).

Here's why:

- The natural logarithm function \( \ln{x} \) is the inverse of the exponential function \( e^x \).
- This means \( e^{\ln{x}} = x \) for any positive \( x \).

Thus, the simplified expression is:

\[
e^{\ln{x}} = x
\]

Would you like a more detailed explanation, or have any questions?

Here are some related questions to expand your understanding:

1. What is the inverse function of \( \ln{x} \)?
2. How does the property of logarithmic and exponential functions relate to solving equations?
3. What is \( e^{\ln{(x^2)}} \) and how does it simplify?
4. How do logarithms with different bases affect the simplification process?
5. Can \( e^{\ln{x}} \) simplify if \( x \) is negative?

**Tip:** Remember, the logarithmic and exponential functions are inverses, which makes simplifications involving \( e^{\ln{x}} \) straightforward!

Understand how to simplify the expression e^ln(x) using properties of exponential and logarithmic functions. Learn key concepts like inverse functions and related formulas. This explanation is suitable for high school students in Grades 10-12.

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