To evaluate the logarithmic expression [tex]\(\ln(\sqrt[7]{e^3})\)[/tex], we can follow these steps:
1. Understanding the Expression: We start with the expression [tex]\(\ln(\sqrt[7]{e^3})\)[/tex]. The term [tex]\(\sqrt[7]{e^3}\)[/tex] represents the 7th root of [tex]\(e^3\)[/tex].
2. Rewrite the Radical Expression: The 7th root of [tex]\(e^3\)[/tex] can be rewritten using exponentiation. In general, the [tex]\(n\)[/tex]-th root of [tex]\(a\)[/tex] can be written as [tex]\(a^{1/n}\)[/tex]. Applying this, we get:
[tex]\[
\sqrt[7]{e^3} = (e^3)^{1/7}
\][/tex]
3. Apply the Law of Exponents: When we have an expression of the form [tex]\((a^m)^n\)[/tex], it can be simplified to [tex]\(a^{m \cdot n}\)[/tex]. Here, [tex]\(m = 3\)[/tex] and [tex]\(n = \frac{1}{7}\)[/tex]. So we get:
[tex]\[
(e^3)^{1/7} = e^{3 \cdot \frac{1}{7}} = e^{3/7}
\][/tex]
4. Use the Property of Logarithms: One of the properties of the natural logarithm is that [tex]\(\ln(a^b) = b \cdot \ln(a)\)[/tex]. Here, [tex]\(a = e\)[/tex] and [tex]\(b = \frac{3}{7}\)[/tex]. Using this property, we have:
[tex]\[
\ln(e^{3/7}) = \frac{3}{7} \cdot \ln(e)
\][/tex]
5. Evaluate the Natural Logarithm: We know that the natural logarithm of [tex]\(e\)[/tex] is 1, i.e., [tex]\(\ln(e) = 1\)[/tex]. Therefore:
[tex]\[
\frac{3}{7} \cdot \ln(e) = \frac{3}{7} \cdot 1 = \frac{3}{7}
\][/tex]
So, the evaluated expression simplifies to:
[tex]\[
\ln(\sqrt[7]{e^3}) = \frac{3}{7}
\][/tex]
Given that [tex]\(\frac{3}{7}\)[/tex] is approximately [tex]\(0.428571428571428\)[/tex], we conclude that:
[tex]\[
\boxed{0.428571428571428}
\][/tex]
