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 $\ln(ab) = \ln(a) + \ln(b)$.