Option C:
[tex]\ln 64 y^{2}[/tex]
Solution:
Given expression: 2 ln 8 + 2 ln y
Applying log rule [tex]a \log b=\log b^{a}[/tex] in the above expression.
⇒ [tex]2 \ln 8+2 \ln y=\ln 8^{2}+\ln y^{2}[/tex]
Applying log rule [tex]\log a+\log b=\log (a b)[/tex] in the above expression.
⇒ [tex]\ln 8^{2}+\ln y^{2}=\ln \left(8^{2} y^{2}\right)[/tex]
We know that [tex]8^{2}=64[/tex]
⇒ [tex]\ln \left(8^{2} y^{2}\right)=\ln 64 y^{2}[/tex]
Hence, the expression in single natural logarithm is [tex]\ln 64 y^{2}[/tex].
