The function given is \( f(x) = \ln(e^x \cdot \cos(x)) \).

To simplify this expression, follow these steps:

1. **Use the logarithm property:**
   \[
   \ln(a \cdot b) = \ln(a) + \ln(b)
   \]
   So,
   \[
   f(x) = \ln(e^x \cdot \cos(x)) = \ln(e^x) + \ln(\cos(x))
   \]

2. **Simplify \( \ln(e^x) \):**
   The natural logarithm of \( e^x \) is just \( x \), because \( \ln(e^x) = x \).

   Therefore,
   \[
   f(x) = x + \ln(\cos(x))
   \]

Thus, the simplified expression for the function is:

\[
f(x) = x + \ln(\cos(x))
\]

Would you like more details on how these steps work? Let me know if you have any questions!

Here are 5 related questions:
1. What is the derivative of \( f(x) = x + \ln(\cos(x)) \)?
2. How do you find the domain of \( f(x) = x + \ln(\cos(x)) \)?
3. Can you evaluate \( f(\pi/4) \) for the function \( f(x) = x + \ln(\cos(x)) \)?
4. What is the range of the function \( f(x) = x + \ln(\cos(x)) \)?
5. How do you find the second derivative of \( f(x) = x + \ln(\cos(x)) \)?

**Tip:** When simplifying logarithmic expressions, always check if logarithm properties can help break down products, quotients, or exponents.

This page provides a step-by-step solution to simplifying the logarithmic expression f(x) = ln(e^x * cos(x)). Learn how to apply logarithmic properties and exponential identities to simplify the function. Suitable for students in Grades 11-12.

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