To simplify the expression $\ln[e(x+7)]$, we can apply logarithmic properties. Here’s the step-by-step breakdown:

Step 1: Use the logarithmic property $\ln(ab) = \ln(a) + \ln(b)$

In this case, the expression inside the logarithm is $e(x + 7)$, which is the product of $e$ and $(x + 7)$. So, we can rewrite this as: $\ln[e(x + 7)] = \ln(e) + \ln(x + 7)$

Step 2: Simplify $\ln(e)$

We know that $\ln(e) = 1$, since the natural logarithm of $e$ is 1. Thus, the expression becomes: $\ln(e) + \ln(x + 7) = 1 + \ln(x + 7)$

Final Answer:

$\ln[e(x + 7)] = 1 + \ln(x + 7)$

This is the simplified form of the expression.

Would you like more details or have any questions?

Here are 5 related questions to deepen your understanding:

1. How would the expression change if it were $\ln[e(x - 7)]$?
2. What happens when you have $\ln[e^{2}(x+7)]$?
3. Can you simplify $\ln[a(x+7)]$ for any constant $a$?
4. What is the logarithmic property for dividing terms inside the logarithm?
5. How do you expand logarithms involving exponents?

Tip: Always remember $\ln(e) = 1$ as it's a basic property of natural logarithms!