To evaluate the expression [tex]\(\ln(e^{\frac{3}{7}})\)[/tex], we can use the properties of logarithms, specifically the natural logarithm. Here's the step-by-step process:
1. Recognize the property of the natural logarithm that we will use: [tex]\(\ln(e^x) = x\)[/tex]. This property states that the natural logarithm of [tex]\(e\)[/tex] raised to any power [tex]\(x\)[/tex] is simply [tex]\(x\)[/tex].
2. Identify the exponent in the expression given. In this case, the exponent is [tex]\(\frac{3}{7}\)[/tex].
3. Apply the property to simplify the expression:
[tex]\[
\ln(e^{\frac{3}{7}}) = \frac{3}{7}
\][/tex]
Therefore, the value of the expression [tex]\(\ln(e^{\frac{3}{7}})\)[/tex] is [tex]\(\frac{3}{7}\)[/tex].
So, [tex]\(\ln(e^{\frac{3}{7}}) = \frac{3}{7}\)[/tex].
When expressed as a decimal, [tex]\(\frac{3}{7} \approx 0.42857142857142855\)[/tex].
