To simplify \( \ln(4e^x) \), we can break it down using logarithmic properties:

### Step-by-step simplification:

\[
\ln(4e^x) = \ln(4) + \ln(e^x)
\]

Using the property \( \ln(e^x) = x \), the equation becomes:

\[
\ln(4e^x) = \ln(4) + x
\]

Thus, the simplified form is:

\[
\ln(4e^x) = \ln(4) + x
\]

Would you like more details or have any questions about this?

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Here are 5 related questions to deepen your understanding:

1. How would you differentiate \( \ln(4e^x) \) with respect to \( x \)?
2. What happens to \( \ln(4e^x) \) if \( x \) approaches infinity?
3. How do the properties of logarithms simplify \( \ln(4e^{2x}) \)?
4. Can you integrate \( \ln(4e^x) \) with respect to \( x \)?
5. What would \( \ln(4e^x) \) be if the base of the logarithm was 10 instead of \( e \)?

**Tip**: When simplifying logarithmic expressions, always look for ways to break them down using properties such as \( \ln(ab) = \ln(a) + \ln(b) \).

Learn how to simplify the expression ln(4e^x) using key logarithmic properties. This problem involves exponential functions and the natural logarithm, making it suitable for students in Grades 10-12.

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