To simplify the given expression [tex]\(\ln[e(x + 3)]\)[/tex], we can use properties of logarithms and exponentials. Here are the steps to reach the solution:
1. Understanding the Properties of Logarithms:
- One of the key properties of logarithms is that the logarithm of a product is the sum of logarithms: [tex]\(\ln(AB) = \ln(A) + \ln(B)\)[/tex].
- Also, the natural logarithm of the base [tex]\(e\)[/tex] is 1: [tex]\(\ln(e) = 1\)[/tex].
2. Break Down the Expression:
- The expression inside the logarithm is a product of [tex]\(e\)[/tex] and [tex]\((x + 3)\)[/tex].
- Using the product property of logarithms, we can separate the logarithm of a product into a sum of two logarithms:
[tex]\[
\ln[e(x + 3)] = \ln[e] + \ln[x + 3].
\][/tex]
3. Simplify the Logarithm:
- We know that [tex]\(\ln(e) = 1\)[/tex].
4. Combine the Results:
- So, we replace [tex]\(\ln(e)\)[/tex] with 1 in our expression:
[tex]\[
\ln[e(x + 3)] = 1 + \ln[x + 3].
\][/tex]
Therefore, the simplified form of [tex]\(\ln[e(x + 3)]\)[/tex] is:
[tex]\[
\boxed{1 + \ln(x + 3)}
\][/tex]
