Answer:
(2).
Step-by-step explanation:
We have the expression:
[tex]\displaystye{\ln(\displaystyle{\frac{\sqrt{e}}{y^3})}[/tex]
First, we can use the difference property of logarithms:
[tex]\ln(x/y)=\ln(x)-\ln(y)[/tex]
Hence, this is equivalent to:
[tex]=\ln(\sqrt{e})-\ln(y^3)[/tex]
We can rewrite the left as:
[tex]=\ln(e^\frac{1}{2})-\ln(y^3)[/tex]
Now, we can use the power property:
[tex]\ln(a^b)=b\ln(a)[/tex]
Essentially, we move the exponent to the front.
Hence, this yields:
[tex]=\frac{1}{2}\ln(e)-3\ln(y)[/tex]
The natural log of e is simply 1. Hence:
[tex]=\frac{1}{2}-3\ln(y)[/tex]
Combine fractions:
[tex]\displaystyle{=\frac{1}{2}-\frac{6\ln(y)}{2} \\ =\frac{1-6\ln(y)}{2}}[/tex]
Hence, our answer is (2).
