Evaluate the following expression. Do not use a calculator.

[tex]\[ \ln e^{\frac{5}{7}} \][/tex]

[tex]\[ \ln e^{\frac{5}{7}} = \square \][/tex] (Type an integer or a fraction.)



Answer :

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]