To solve the equation \( e^{x+3} = 5 \), we can follow these steps:

1. **Take the natural logarithm (ln) of both sides** to eliminate the exponential:
   \[
   \ln(e^{x+3}) = \ln(5)
   \]

2. **Simplify the left side** using the property \( \ln(e^a) = a \):
   \[
   x + 3 = \ln(5)
   \]

3. **Solve for \( x \)** by isolating it:
   \[
   x = \ln(5) - 3
   \]

4. **Calculate the value of \( \ln(5) \)** (using a calculator):
   \[
   \ln(5) \approx 1.6094
   \]
   So,
   \[
   x \approx 1.6094 - 3 = -1.3906
   \]

Thus, the solution is approximately \( x \approx -1.3906 \).

Would you like further details or have any questions?

Here are 5 related questions:
1. How would you solve \( e^{x+2} = 10 \)?
2. What is the natural logarithm, and how does it relate to the exponential function?
3. Can you solve \( e^{2x} = 7 \) for \( x \)?
4. How do we simplify expressions involving exponents and logarithms?
5. What happens if the base of the exponential is not \( e \)?

**Tip**: Always remember that \( \ln(e^x) = x \) when solving logarithmic and exponential equations!

Explore how to solve the exponential equation e^{x+3} = 5 using natural logarithms. This detailed step-by-step solution covers essential concepts in exponential functions and logarithms, making it ideal for students in Grades 10-12.

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