The expression in the image is:

$\ln(e^3) + \ln(e^2)$

To solve this, let's break it down step by step:

Step 1: Simplify each logarithm

Using the logarithm property $\ln(a^b) = b \cdot \ln(a)$:

1. $\ln(e^3) = 3 \cdot \ln(e)$
2. $\ln(e^2) = 2 \cdot \ln(e)$

Step 2: Recognize that $\ln(e) = 1$

Since the natural logarithm of $e$ is 1 ($\ln(e) = 1$):

1. $3 \cdot \ln(e) = 3 \cdot 1 = 3$
2. $2 \cdot \ln(e) = 2 \cdot 1 = 2$

Step 3: Add the simplified expressions

Now, sum the simplified values:

$3 + 2 = 5$

Final Answer:

$\ln(e^3) + \ln(e^2) = 5$

Would you like more details or have any questions?

Here are some related questions to explore:

1. What is the derivative of the function $\ln(x)$?
2. How do logarithm properties simplify expressions involving multiplication or division?
3. What is the integral of $\ln(x)$?
4. Can you solve $\ln(e^x) + \ln(e^y)$ for arbitrary $x$ and $y$?
5. What is the natural logarithm's behavior as $x$ approaches infinity?

Tip: When working with logarithms, remember to use properties like the power rule ($\ln(a^b) = b \cdot \ln(a)$) and the product rule ($\ln(ab) = \ln(a) + \ln(b)$) to simplify expressions efficiently.