b) ${\log }_{\frac{\sqrt{3}}{3}}\left(81\sqrt[3]{3}\right)$

Oznaczmy ${\log }_{\frac{\sqrt{3}}{3}}\left(81\sqrt[3]{3}\right)=c$. Z definicji logarytmu mamy:

${\left(\frac{\sqrt{3}}{3}\right)}^c=81\sqrt[3]{3}$

${\left(\frac{3^{\frac{1}{2}}}{3^1}\right)}^c=3^4\cdot 3^{\frac{1}{3}}$

${\left(3^{\frac{1}{2}-1}\right)}^c=3^{4+\frac{1}{3}}$

$3^{-\frac{1}{2}c}=3^{\frac{13}{3}}$

$-\frac{1}{2}c=\frac{13}{3}|\cdot \left(-2\right)$

$c=-\frac{26}{3}$

Odp.: ${\log }_{\frac{\sqrt{3}}{3}}\left(81\sqrt[3]{3}\right)=-\frac{26}{3}$