Use the quotient property of logarithms to write the logarithm as a difference of logarithms.

[tex]\[ \ln \left(\frac{e}{5}\right) \][/tex]



Answer :

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].