[latex]log_ba = sqrt{3} [/latex] [latex]log_{frac{sqrt{a}}{b}} frac{sqrt[3]{a}}{sqrt{b}} = frac{log_b frac{sqrt[3]{a}}{sqrt{b}}}{log_b frac{sqrt{a}}{b}}} = frac{log_b sqrt[3]{a}- log_b sqrt{b}}{log_b sqrt{a} - log_b b} = frac{log_b a^{frac{1}{3}}- log_b b^{frac{1}{2}}}{log_ba^{frac{1}{2}} - 1} =[/latex] [latex]=frac{frac{1}{3} cdot log_b a-frac{1}{2} cdot log_b b}{frac{1}{2} cdot log_ba - 1} = frac{frac{1}{3} cdot sqrt{3} -frac{1}{2} cdot 1}{frac{1}{2} cdot sqrt{3} - 1} = frac{frac{2cdot sqrt{3}}{6} -frac{3}{6}}{frac{3cdot sqrt{3}}{6} - frac{6}{6}} = frac{frac{2sqrt{3}-3}{6}}{frac{3sqrt{3}-6}{6}} =frac{2sqrt{3}-3}{6} : frac{3sqrt{3}-6}{6}=[/latex] [latex]=frac{2sqrt{3}-3}{6} cdot frac{6}{3sqrt{3}-6} =frac{2sqrt{3}-3}{3sqrt{3}-6} =frac{(2sqrt{3}-3)(sqrt{3}+2)}{3 cdot (sqrt{3}-2)(sqrt{3}+2)} =frac{(2sqrt{3}-3)(sqrt{3}+2)}{3 cdot (3-4)} =[/latex] [latex]=frac{6 +4sqrt{3}-3sqrt{3}-6}{3 cdot (-1)} = frac{sqrt{3}}{-3} = -frac{sqrt{3}}{3}[/latex]
