To find the domain of the function \( f(x) = 1 - \ln(x) \), we need to consider the properties of the natural logarithm function, \( \ln(x) \).

### Key property of the natural logarithm:
- The natural logarithm \( \ln(x) \) is only defined for positive real numbers. In other words, \( \ln(x) \) is defined if and only if \( x > 0 \).

Thus, the expression \( \ln(x) \) in \( 1 - \ln(x) \) implies that \( x \) must be greater than 0.

### Domain of \( f(x) \):
The domain of \( f(x) = 1 - \ln(x) \) is:

\[
x > 0
\]

In interval notation, the domain is:
\[
(0, \infty)
\]

### Summary:
The function \( 1 - \ln(x) \) is defined for all \( x \) in the interval \( (0, \infty) \).

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Would you like more details on the logarithmic function's properties, or do you have any questions?

### Related Questions:
1. What is the range of the function \( f(x) = 1 - \ln(x) \)?
2. How would the domain change if the function was \( 1 + \ln(x) \)?
3. What happens to the graph of \( f(x) = 1 - \ln(x) \) as \( x \) approaches infinity?
4. How do you find the inverse of the function \( f(x) = 1 - \ln(x) \)?
5. What is the derivative of \( f(x) = 1 - \ln(x) \)?

#### Tip: 
Always remember that the natural logarithm is undefined for non-positive values of \( x \). So the domain will always exclude zero and negative numbers.

Learn how to find the domain of the logarithmic function f(x) = 1 - ln(x). This step-by-step guide explains the key property of logarithmic functions, focusing on the domain restrictions for ln(x) being defined only when x > 0. Suitable for students in Grades 10-12.

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