The equation given in the image is:

$\ln(x + 3) - 5 = -2$

Let's solve this step by step.

Step 1: Isolate the logarithmic term

First, add 5 to both sides of the equation to get rid of the -5 on the left side:

$\ln(x + 3) = -2 + 5$ $\ln(x + 3) = 3$

Step 2: Exponentiate both sides

Now, exponentiate both sides using the base $e$, which is the natural base for logarithms:

$x + 3 = e^3$

Step 3: Solve for $x$

Subtract 3 from both sides to isolate $x$:

$x = e^3 - 3$

The value of $e^3$ (approximated to 4 decimal places) is:

$e^3 \approx 20.0855$

So,

$x \approx 20.0855 - 3 = 17.0855$

Thus, the solution is approximately:

$x \approx 17.09$

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

Here are 5 related questions that might expand your understanding:

1. How do we handle different types of logarithms (e.g., natural logarithms vs. base-10 logarithms)?
2. What happens if the equation had a base other than $e$?
3. Can logarithms be negative, and what does that imply in an equation like this?
4. How would you solve the equation if instead of a constant on the right-hand side, you had another logarithmic expression?
5. Why is it important to check the domain of a logarithmic function?

Tip: Always remember that the argument of a logarithmic function (in this case, $x + 3$) must be positive for the logarithm to be defined. This helps avoid invalid solutions.