https://file.aiquickdraw.com/mathbot/file/1728498660709y6gh3n76.jpg

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.

Learn how to solve the logarithmic equation ln(x + 3) - 5 = -2 step by step. This problem covers key concepts in algebra, logarithmic functions, and exponentiation. Suitable for high school students in Grades 10-12.

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