To evaluate the expression [tex]\(\ln(e^{\frac{5}{7}})\)[/tex], we can use the properties of logarithms. Specifically, one useful property of the natural logarithm is:
[tex]\[
\ln(e^a) = a
\][/tex]
This property tells us that the natural logarithm of [tex]\(e\)[/tex] raised to any power [tex]\(a\)[/tex] is simply [tex]\(a\)[/tex]. It applies directly to our given expression.
Given the expression [tex]\(\ln(e^{\frac{5}{7}})\)[/tex], let's apply this property:
[tex]\[
\ln(e^{\frac{5}{7}}) = \frac{5}{7}
\][/tex]
This is because the exponent [tex]\(\frac{5}{7}\)[/tex] comes down as the result of applying the natural logarithm to [tex]\(e\)[/tex] raised to that power.
So, we have:
[tex]\[
\ln(e^{\frac{5}{7}}) = \frac{5}{7}
\][/tex]
Therefore, the evaluated value of [tex]\(\ln(e^{\frac{5}{7}})\)[/tex] is:
[tex]\[
0.7142857142857143
\][/tex]
Thus,
[tex]\[
\ln(e^{\frac{5}{7}}) = \frac{5}{7}
\][/tex]
or approximately:
[tex]\[
0.7142857142857143
\][/tex]
