a)
b=4\sqrt2=2^2*2^{\frac{1}{2}}=2^{4+1}{2}=2^{\frac{5}{2}}
log_22^{\frac{5}{2}}=\frac{5}{2}
b)
b=16\sqrt[3]{2}=2^4*2^{\frac{1}{3}}=2^{\frac{12+1}{3}}=2^{\frac{13}{3}}
log_22^{\frac{13}{3}}=\frac{13}{3}
c)
b=\frac{\sqrt2}{8}=\frac{2^{\frac{1}{2}}}{2^3}=2^{\frac{1}{2}-3}=2^{\frac{1-6}{2}}=2^{-\frac{5}{2}}
log_22^{-\frac{5}{2}}=-\frac{5}{2}
d)
b=\frac{\sqrt[3]{2}}{32}=\frac{2^{\frac{1}{3}}}{2^5}=2^{\frac{1}{3}-5}=2^{\frac{1-15}{3}}=2^{-\frac{14}{3}}
log_22^{-\frac{14}{3}}=-\frac{14}{3}