zadanie 1 a) [latex]log_7 sqrt[3]{49} = x\ 7^x = sqrt[3]{49}\ 7^x = sqrt[3]{7^2} \ 7^x = 7^{frac{2}{3}} \ x=frac{2}{3}[/latex] b) [latex]log_6 900 -2log_6 5 = log_6 900 - log_6 5^2 = \ =log_6 900 - log_6 25 = log_6 (900 :25) = log_6 26 = 2 bo 6^2 = 36[/latex] c) [latex]36^{1 + log_6 3} = 36^1 cdot 36^{log_6 3} = \ =36 cdot 6^{2log_6 3} = 36cdot 6^{log_6 3^2}=\ =36cdot 6^{log_6 9} = 36cdot 9 = 324[/latex] d) [latex]log_8 2sqrt[3]4 = x\ 8^x = 2sqrt[3]4 \ (2^3)^x = 2^1 cdot 2^{frac{2}{3}} \ 2^{3x} = 2^{1frac{2}{3}} \ 3x = 1frac{2}{3}\ 3x = frac{5}{3}\ x=frac{5}{9}[/latex] zadanie 2 a) [latex]frac{ sqrt{32}}{sqrt[3]{4sqrt2}} = frac{ sqrt{2^5}}{sqrt[3]{ 4 cdot 2^{frac{1}{2}}}} =\ =frac{ (2^5)^{frac{1}{2}}}{sqrt[3]{ 2^2 cdot 2^{frac{1}{2}}}} = frac{2^{frac{5}{2}}}{sqrt[3]{2^{frac{5}{2}}}}=\ =frac{2^{frac{5}{2}}}{2^{frac{5}{6}}}= 2^{frac{5}{2} - frac{5}{6}} = 2^{frac{10}{6}} =2^{frac{5}{3}}[/latex] b) [latex]frac{3^{10} +3^9 + 5cdot 3^8}{17cdot 81^2} =\ =frac{3^8 (3^2 + 3^1 + 5)}{17 cdot (3^4)^2} =\ =frac{3^8 (9+3+5)}{17cdot 3^8} = \ =frac{17}{17} = 1[/latex]
