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Let's solve the given inequalities:

### (a) \(\ln(x) < 0\)
To solve this inequality, we need to determine for what values of \(x\), the natural logarithm of \(x\) is less than zero.

1. Recall that the natural logarithm function \(\ln(x)\) is less than zero when \(0 < x < 1\), because the graph of \(\ln(x)\) crosses the x-axis at \(x = 1\), and for \(x > 1\), \(\ln(x)\) is positive.

Thus, the solution to the inequality \(\ln(x) < 0\) is:
\[
(0, 1)
\]
in interval notation.

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### (b) \(e^x > 3\)
To solve this inequality, follow these steps:

1. Take the natural logarithm of both sides of the inequality:
   \[
   \ln(e^x) > \ln(3)
   \]
2. Simplifying the left-hand side using the property \(\ln(e^x) = x\), we get:
   \[
   x > \ln(3)
   \]
3. Since \(\ln(3)\) is a constant (approximately 1.0986), the solution is:
\[
x > \ln(3)
\]
In interval notation, this is written as:
\[
(\ln(3), \infty)
\]

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Let me know if you'd like further details or explanations! Here are some related questions you can explore:

1. How do you graph the function \( \ln(x) \)?
2. Can \( \ln(x) \) ever be negative for values greater than 1?
3. What happens to \( e^x \) as \( x \to -\infty \)?
4. How would you solve \( e^x \geq 3 \) instead of \( e^x > 3 \)?
5. What is the general method for solving exponential inequalities?

**Tip:** For exponential inequalities like \( e^x > a \), you can always take the natural logarithm of both sides to simplify the inequality.

Solve the inequalities (a) ln(x) < 0 and (b) e^x > 3 in interval notation.

Learn how to solve logarithmic and exponential inequalities, such as ln(x) < 0 and e^x > 3, using interval notation. This problem focuses on understanding the properties of logarithms and exponentials, suitable for high school students in grades 10-12.

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