[latex]1.Oblicz \ a) log_{3} frac{27}{ sqrt[4]{243}} \ d) log_{5} frac{25 sqrt[3]{5}} {sqrt[4]{125}} \ [/latex]

[latex]a) log_{3}frac{27}{sqrt[4]{243}}=log_{3}frac{3^3}{(3^5)^{frac{1}{4}}}=log_{3}3^{frac{7}{4}}=frac{7}{4}\ \ b) log_{5}frac{25sqrt[3]{5}}{sqrt[4]{125}}=log_{5}frac{5^2cdot 5^{frac{1}{3}}}{(5^3)^{frac{1}{4}}}=log_{5}frac{5^{frac{7}{3}}}{5^{frac{3}{4}}}=log_{5}5^{frac{19}{12}}=frac{19}{12}[/latex]

Korzystamy z definicji logarytmu: [latex]log_{a}b=c iff a^{c}=b[/latex] a) [latex]log_{3}frac{27}{sqrt[4]{243}}=x\\3^{x}=frac{27}{sqrt[4]{243}}\\3^{x}=frac{3^{3}}{((3^{5})^{frac{1}{4}}}\\3^{x}=frac{3^{3}}{3^{frac{5}{4}}}\\3^{x}=3^{3-frac{5}{4}}\\3^{x}=3^{frac{12}{4}-frac{5}{4}}\\3^{x}=3^{frac{7}{4}}\\x=frac{7}{4}\x=1frac{3}{4} o underline{underline{log_{3}frac{27}{sqrt[4]{243}}=1frac{3}{4}}}[/latex] b) [latex]log_{5}frac{25sqrt[3]{5}}{sqrt[4]{125}}=x\\5^{x}=frac{25sqrt[3]{5}}{sqrt[4]{125}}\\5^{x}=frac{5^{2}*5^{frac{1}{3}}}{(5^{3})^{frac{1}{4}}}[/latex] [latex]5^{x}=frac{5^{frac{7}{3}}}{5^{frac{3}{4}}}\\5^{x}=5^{frac{7}{3}-frac{3}{4}}\\5^{x}=5^{frac{28}{12}-frac{9}{12}}\\5^{x}=5^{frac{19}{12}}\\x=frac{19}{12} \\x=1frac{7}{12} o underline{underline{log_{5}frac{25sqrt[3]{5}}{sqrt[4]{125}}=1frac{7}{12}}}[/latex]