Korzystamy z definicji logarytmu: [latex]log_{a}b=c iff a^{c}=b[/latex] a) [latex]log_{frac{2}{3}}frac{81}{16}=x\\(frac{2}{3})^{x}=frac{81}{16}\\(frac{2}{3})^{x}=(frac{16}{81})^{-1}\\(frac{2}{3})^{x}=((frac{2}{3})^{4})^{-1}\\(frac{2}{3})^{x}=(frac{2}{3})^{-4}\\x=-4 o underline{log_{frac{2}{3}}=frac{81}{16}=-4}[/latex] b) [latex]log_{frac{1}{3}}x=-frac{1}{2}\\x=(frac{1}{3})^{-frac{1}{2}}\\x=3^{frac{1}{2}}\\x=sqrt{3} o underline{log_{frac{1}{3}}sqrt{3}=-frac{1}{2}}[/latex] c) [latex]log_{2}x=-frac{2}{3}\\x=2^{-frac{2}{3}}\\x=(frac{1}{2})^{frac{2}{3}}\\x=sqrt[3]{(frac{1}{2})^{2}}\\x=sqrt[3]{frac{1}{4}} \\x=frac{sqrt[3]{1}}{sqrt[3]{4}}*frac{sqrt[3]{4}}{sqrt[3]{4}}*frac{sqrt[3]{4}}{sqrt[3]{4}}=frac{1*sqrt[3]{16}}{sqrt[3]{64}}=frac{sqrt[3]{8*2}}{4}=frac{2sqrt[3]{2}}{4}=frac{sqrt[3]{2}}{2}\\ o underline{log_{2}frac{sqrt[3]{2}}{2}=-frac{2}{3}}[/latex]
