https://file.aiquickdraw.com/mathbot/file/1726162184233f4he6zv8.jpg

The inequality shown in the image is:

\[
\ln(x) > 3
\]

This is a logarithmic inequality. To solve it, we follow these steps:

### 1. Rewrite the inequality:
The natural logarithm (ln) is the logarithm to the base \(e\). To eliminate the logarithm, we can exponentiate both sides of the inequality with base \(e\), because the exponential function is the inverse of the natural logarithm.

\[
e^{\ln(x)} > e^3
\]

This simplifies to:

\[
x > e^3
\]

### 2. Calculate \(e^3\):
Using an approximation, we get:

\[
e^3 \approx 20.0855
\]

Thus, the solution to the inequality is:

\[
x > 20.0855
\]

This means that for the inequality \(\ln(x) > 3\) to hold, \(x\) must be greater than approximately 20.0855.

Would you like any further details or have any questions?

Here are 5 related questions for further exploration:
1. How do you solve logarithmic inequalities with different bases?
2. What is the domain of the natural logarithmic function?
3. How do you graph the inequality \(\ln(x) > 3\)?
4. Can logarithmic inequalities have multiple solutions?
5. What are the applications of logarithmic inequalities in real life?

**Tip**: When solving logarithmic inequalities, always remember to check the domain of the logarithmic function! The argument of the logarithm must be positive.

Learn how to solve the logarithmic inequality ln(x) > 3 using logarithmic properties and exponential functions. The solution shows that x must be greater than approximately 20.0855. This explanation is ideal for students in Grades 9-12 learning about logarithmic inequalities.

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