\log _{2} 8=3
\log _{3}\left(\cfrac{1}{27}\right)=-3
\log _{\frac{1}{5}} 25=-2
\log _{4} 2=\cfrac{1}{2}
a^{\log _{ a } b }= b a{n}
\log _{a} a=1
\log _{a} 1=0
\log _{a} a^{c}=c
\log _{\sqrt{2}}(2 \sqrt{8})=
\log _{\sqrt{ 2 }}\Big((\sqrt{2})^{ 2 } \cdot(\sqrt{2})^{ 3 }\Big)=
\log _{\sqrt{ 2 }}(\sqrt{2})^{ 2 +3}= 2 +3= 5 ;
\log _{\frac{1}{15}}(225 \sqrt[3]{15})=
\log _{\frac{ 1 }{15}}\Big( 15 ^{2} \cdot 15 ^{\frac{1}{ 3 }}\Big)=
\log _{\frac{ 1 }{15}}\Big(\cfrac{ 1 }{15}\Big)^{-( 2 +\frac{1}{ 3 })}=
-2-\cfrac{ 1 }{3}=- 2 \cfrac{1}{ 3 } ;
\log _{\sqrt{3}} 81 \sqrt{3}=
\log _{\sqrt{ 3 }}\Big((\sqrt{3})^{ 2 \cdot 4} \cdot \sqrt{ 3 }\Big)=
\log _{\sqrt{ 3 }}(\sqrt{3})^{ 2 \cdot 4+ 1 }=
2 \cdot 4 +1= 9