https://file.aiquickdraw.com/mathbot/file/1726162690874k4vd0jup.jpg

The image shows the inequality:

\[
\ln(x) < 1
\]

To solve for \(x\), we need to find the values of \(x\) that satisfy this inequality. Here's the step-by-step solution:

1. **Rewrite the Inequality**:
   \[
   \ln(x) < 1
   \]

2. **Exponentiate Both Sides**:
   To eliminate the natural logarithm, we exponentiate both sides with the base \(e\):
   \[
   e^{\ln(x)} < e^1
   \]

   Since \(e^{\ln(x)} = x\), we get:
   \[
   x < e
   \]

3. **Consider the Domain of the Natural Logarithm**:
   The natural logarithm \(\ln(x)\) is defined only for \(x > 0\). Thus, the solution must satisfy both \(x > 0\) and \(x < e\).

4. **Final Solution**:
   \[
   0 < x < e
   \]

Therefore, the solution to the inequality \(\ln(x) < 1\) is \(x \in (0, e)\).

Would you like more details on this solution or have any questions?

### Related Questions:

1. What is the value of \(e\) and how is it approximated?
2. How would the inequality change if it were \(\ln(x) > 1\)?
3. What is the domain of the natural logarithm function \(\ln(x)\)?
4. How does the natural logarithm behave as \(x\) approaches 0 from the positive side?
5. What is the derivative of the natural logarithm function?

**Tip:** When solving inequalities involving logarithmic functions, always consider the domain of the function and the behavior of the logarithm near its boundaries.

Learn how to solve the logarithmic inequality ln(x) < 1 using properties of logarithms and exponential functions. The solution involves exponentiating both sides and considering the domain of the natural logarithm function. Suitable for students in Grades 11-12.

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