Definicja logarytmu: [latex]log_{a}c=b <=> a^{c}=b[/latex] --------------------------------------------------------------- [latex]a) log_{4}8sqrt[4]{2}=c <=> 4^{c}=8sqrt[4]{2}\ 4^{c}=8sqrt[4]{2}\ (2^{2})^{c}=2^{3}*2^{frac{1}{4}}\ 2^{2*c}=2^{3+frac{1}{4}}\ 2^{2c}=2^{3frac{1}{4}}[/latex] [Podstawy potęgo po obu stronach są równe, przechodzę więc do działań na wykładnikach potęg] [latex]2c=3frac{1}{4} |*frac{1}{2}\ c=frac{13}{4}*frac{1}{2}\ c=frac{13}{8}[/latex] ------------------------------------------------------ [latex]log_{frac{1}{5}}25sqrt{5}=c <=> (frac{1}{5})^{c}=25sqrt{5}\ (frac{1}{5})^{c}=25sqrt{5}\ (5^{-1})^{c}=5^{2}*5^{frac{1}{2}}\ 5^{-1*c}=5^{2+frac{1}{2}}\ 5^{-c}=5^{2frac{1}{2}}\ -c=2frac{1}{2} |*(-1)\ c=-2frac{1}{2}[/latex] ------------------------------------------------------ [latex]log_{sqrt{3}}sqrt[3]{9}=c <=> sqrt{3}^{c}=sqrt[3]{9}\ sqrt{3}^{c}=sqrt[3]{9}\ (3^{frac{1}{2}})^{c}=9^{frac{1}{3}}\ 3^{frac{1}{2}*c}=(3^{2})^{frac{1}{3}}\ 3^{frac{1}{2}c}=3^{2*frac{1}{3}}\ 3^{frac{1}{2}c}=3^{2frac{1}{3}}\ frac{1}{2}c=2frac{1}{3} |*2\ c=frac{7}{3}*2\ c=frac{14}{3}[/latex]
