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!