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The given expression is:

\[
\exp[\ln(x)] - \ln[\exp(x)]
\]

### Step-by-step Simplification:

1. **Simplify \(\exp[\ln(x)]\):**
   - The exponential function \(\exp(y)\) is the inverse of the natural logarithm \(\ln(y)\).
   - Thus, \(\exp[\ln(x)] = x\) for \(x > 0\).

2. **Simplify \(\ln[\exp(x)]\):**
   - Similarly, since \(\ln[\exp(x)] = x\), the second part simplifies to \(x\).

Now the expression becomes:

\[
x - x = 0
\]

So, the simplified form of the expression is \(0\).

### Domain of the function:

- The domain is determined by the conditions under which the original expression is valid. Specifically:
   - For \(\ln(x)\) to be defined, \(x > 0\).
   - For \(\exp(x)\), there are no restrictions because the exponential function is defined for all real numbers.

Thus, the domain of the function is:

\[
x > 0
\]

### Summary:

- The simplified expression is \(0\).
- The domain of the function is \(x > 0\).

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Would you like a detailed explanation or have any further questions? 

Here are some related questions you could explore:

1. What is the general relationship between logarithmic and exponential functions?
2. How does the inverse property of \(\ln(x)\) and \(\exp(x)\) simplify expressions?
3. How can the domain of logarithmic functions be determined?
4. What other functions have similar inverse relationships like \(\ln\) and \(\exp\)?
5. What are some applications of logarithmic and exponential functions in real-world scenarios?

**Tip:** Remember, the domain of a logarithmic function is always restricted to positive real numbers because \(\ln(x)\) is undefined for \(x \leq 0\).

Simplify the following expression: exp[ln(x)] - ln[exp(x)]. Indicate also the domain of the function associated with the corresponding expression.

This page explains how to simplify the expression exp[ln(x)] - ln[exp(x)] step by step. It covers key mathematical concepts, including the inverse properties of exponential and logarithmic functions. The domain of the function is also discussed, making it suitable for students in Grades 11-12.

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