Sure! Let's solve the problem step by step using the quotient property of logarithms.
Given expression:
[tex]\[
\ln \left(\frac{e}{5}\right)
\][/tex]
The quotient property of logarithms states that:
[tex]\[
\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)
\][/tex]
In this case, [tex]\( a = e \)[/tex] and [tex]\( b = 5 \)[/tex]. Applying the quotient property, we get:
[tex]\[
\ln \left(\frac{e}{5}\right) = \ln(e) - \ln(5)
\][/tex]
Now, evaluate each term separately.
1. The natural logarithm of [tex]\( e \)[/tex] (since [tex]\( e \)[/tex] is the base of the natural logarithm):
[tex]\[
\ln(e) = 1
\][/tex]
2. The natural logarithm of [tex]\( 5 \)[/tex] is approximately:
[tex]\[
\ln(5) \approx 1.6094379124341003
\][/tex]
Now, substitute these values back into the equation:
[tex]\[
\ln \left(\frac{e}{5}\right) = 1 - 1.6094379124341003
\][/tex]
Perform the subtraction:
[tex]\[
1 - 1.6094379124341003 \approx -0.6094379124341003
\][/tex]
So, the final result is:
[tex]\[
\ln \left(\frac{e}{5}\right) \approx -0.6094379124341003
\][/tex]
Hence, the logarithm [tex]\(\ln \left(\frac{e}{5}\right)\)[/tex] can be written as the difference of logarithms, resulting in approximately [tex]\(-0.6094379124341003\)[/tex].
