To simplify [tex]\(\ln (e x)\)[/tex], we can use the properties of logarithms. One essential property is:
[tex]\[
\ln(a \cdot b) = \ln(a) + \ln(b)
\][/tex]
So, in the given expression [tex]\(\ln (e x)\)[/tex], we can decompose it as follows:
1. Identify the components inside the logarithm. Here [tex]\(a = e\)[/tex] and [tex]\(b = x\)[/tex].
2. Apply the logarithm product rule:
[tex]\[
\ln (e x) = \ln(e) + \ln(x)
\][/tex]
3. Recognize that [tex]\(\ln(e) = 1\)[/tex] because [tex]\(e\)[/tex] is the base of the natural logarithm:
[tex]\[
\ln(e) = 1
\][/tex]
4. Substitute [tex]\(\ln(e)\)[/tex] with 1 in the equation:
[tex]\[
\ln(e x) = 1 + \ln(x)
\][/tex]
Therefore, the expression [tex]\(\ln (e x)\)[/tex] simplifies to:
[tex]\[
1 + \ln(x)
\][/tex]
