Zad. 3 a) [latex]log_{sqrt{5}} 5sqrt[3]{5}= x[/latex] [latex](sqrt{5})^x= 5sqrt[3]{5}[/latex] [latex](5^{frac{1}{2}})^x= 5 cdot 5^{frac{1}{3}}[/latex] [latex]5^{frac{1}{2}x}= 5^{1+frac{1}{3}}[/latex] [latex]frac{1}{2}x= 1+frac{1}{3}[/latex] [latex]frac{1}{2}x= frac{4}{3} / cdot 2[/latex] [latex]x= frac{8}{3}[/latex] [latex]x= 2frac{2}{3}[/latex] b) [latex]log_{sqrt[3]{3}} 27= x[/latex] [latex](sqrt[3]{3}})^x = 27[/latex] [latex](3^{frac{1}{3}})^x = 3^3[/latex] [latex]3^{frac{1}{3}x} = 3^3[/latex] [latex]frac{1}{3}x = 3 / cdot 3[/latex] [latex]x = 9[/latex] c) [latex]log_2 8sqrt{2} =x[/latex] [latex]2^x= 8sqrt{2}[/latex] [latex]2^x= 2^3 cdot 2^{frac{1}{2}}[/latex] [latex]2^x= 2^{3 +frac{1}{2}}[/latex] [latex]x = 3 +frac{1}{2}[/latex] [latex]x = 3frac{1}{2}[/latex] d) [latex]log_{frac{1}{3}} 81sqrt{3} = x[/latex] [latex](frac{1}{3})^x= 81sqrt{3}[/latex] [latex](3^{-1})^x= 3^4 cdot 3^{frac{1}{2}}[/latex] [latex]3^{-x} = 3^{4+frac{1}{2}}[/latex] [latex]-x = 4+frac{1}{2}[/latex] [latex]-x = 4frac{1}{2} / cdot (- 1)[/latex] [latex]x = - 4frac{1}{2}[/latex] e) [latex]log_4 8sqrt[4]{2}=x[/latex] [latex]4^x = 8sqrt[4]{2}[/latex] [latex](2^2)^x = 2^3 cdot 2^{frac{1}{4}}[/latex] [latex]2^{2x}= 2^{3+frac{1}{4}}[/latex] [latex]2x= 3+frac{1}{4}[/latex] [latex]2x= 3frac{1}{4}[/latex] [latex]2x= frac{13}{4} /:2[/latex] [latex]x= frac{13}{8} [/latex] [latex]x= 1frac{5}{8} [/latex] f) [latex]log_{frac{1}{5}} 25sqrt{5} = x[/latex] [latex](frac{1}{5})^x=25sqrt{5}[/latex] [latex](5^{-1})^x=5^2 cdot 5^{frac{1}{2}}[/latex] [latex]5^{-x}= 5^{2+frac{1}{2}}[/latex] [latex]-x= 2+frac{1}{2}[/latex] [latex]-x= 2frac{1}{2} / cdot (- 1)[/latex] [latex]x= -2frac{1}{2}[/latex] g) [latex]log_{sqrt{3}} sqrt[3]{9}=x[/latex] [latex](sqrt{3})^x = sqrt[3]{9}[/latex] [latex](3^{frac{1}{2}})^x = 9^{frac{1}{3}}[/latex] [latex]3^{frac{1}{2}x}= (3^2)^{frac{1}{3}}[/latex] [latex]3^{frac{1}{2}x}= 3^{frac{2}{3}}[/latex] [latex]frac{1}{2}x= frac{2}{3} / cdot 2[/latex] [latex]x= frac{4}{3}[/latex] [latex]x= 1frac{1}{3}[/latex] h) [latex]log_{2sqrt{2}} 4sqrt{8}=x[/latex] [latex](2sqrt{2})^x= 4sqrt{8}[/latex] [latex](2 cdot 2^{frac{1}{2}})^x= 4 cdot 8^{frac{1}{2}}[/latex] [latex](2^{1+frac{1}{2}})^x= 2^2 cdot (2^3)^{frac{1}{2}}[/latex] [latex](2^{frac{3}{2}})^x= 2^2 cdot 2^{frac{3}{2}}[/latex] [latex]2^{frac{3}{2}x}= 2^{2+frac{3}{2}}[/latex] [latex]frac{3}{2}x= 2+frac{3}{2}[/latex] [latex]frac{3}{2}x= 2frac{3}{2}[/latex] [latex]frac{3}{2}x= frac{7}{2} / : (frac{3}{2})[/latex] [latex]x= frac{7}{3}[/latex] [latex]x= 2frac{1}{3}[/latex] Zad. 4 a) [latex]log_3 x = -1[/latex] [latex]3^{-1} = x[/latex] [latex]x = frac{1}{3}[/latex] b) [latex]log_5 x = 3[/latex] [latex]5^3 = x[/latex] [latex]x = 125[/latex] c) [latex]log_{frac{1}{2}} x = - 2[/latex] [latex](frac{1}{2})^{-2} = x[/latex] [latex]x = 2^2[/latex] [latex]x = 4[/latex] d) [latex]log_{frac{1}{2}} x = -frac{1}{2}}[/latex] [latex](frac{1}{2})^{-frac{1}{2}} = x[/latex] [latex]x = 2^{frac{1}{2}}[/latex] [latex]x = sqrt{2}[/latex] e) [latex]log_2 x = -frac{2}{3}[/latex] [latex]2^{-frac{2}{3}} = x[/latex] [latex]x = 2^{-1} cdot 2^{frac{1}{3}}[/latex] [latex]x = frac{1}{2} cdot sqrt[3]{2}[/latex] [latex]x = frac{sqrt[3]{2}}{2}[/latex] f) [latex]log_4 x = 0[/latex] [latex]4^0 = x[/latex] [latex]x = 1[/latex] g) [latex]log_2 x = 10[/latex] [latex]2^{10} = x[/latex] [latex]x = 1024[/latex] h) [latex]log_{2sqrt{2}} x = - 3[/latex] [latex](2sqrt{2})^{-3} = x[/latex] [latex]x = frac{1}{(2sqrt{2})^3}[/latex] [latex]x = frac{1}{16sqrt{2}}[/latex] [latex]x = frac{sqrt{2}}{16sqrt{2} cdot sqrt{2}}[/latex] [latex]x = frac{sqrt{2}}{32}[/latex] Zad. 5 a) [latex]log_x 25 = 2[/latex] [latex]x^2 = 25[/latex] [latex]x^2 = 5^2[/latex] [latex]x = 5[/latex] b) [latex]log_x 81 = 4[/latex] [latex]x^4 = 81[/latex] [latex]x^4 = 3^4[/latex] [latex]x = 3[/latex] c) [latex]log_x 3 = frac{1}{2}[/latex] [latex]x^{frac{1}{2}} = 3[/latex] [latex]x^{frac{1}{2}} = 9^{frac{1}{2}} [/latex] [latex]x = 9[/latex] d) [latex]log_x 2 = frac{1}{3}[/latex] [latex]x^{frac{1}{3}} = 2[/latex] [latex]x^{frac{1}{3}} =8^{frac{1}{3}}[/latex] [latex]x = 8[/latex] e) [latex]log_x frac{1}{9} = - 2[/latex] [latex]x^{-2} = frac{1}{9}[/latex] [latex]x^{-2} = 3^{-2}[/latex] x = [latex]x = 3[/latex] f) [latex]log_x 64 = - 3[/latex] [latex]x^{-3} = 64[/latex] [latex]x^{-3} = (frac{1}{4})^{-3}[/latex] [latex]x = frac{1}{4}[/latex] g) [latex]log_x 27 = 3[/latex] [latex]x^3 = 27[/latex] [latex]x^3 = 3^3[/latex] [latex]x = 3[/latex] h) [latex]log_x 36 = - 2[/latex] [latex]x^{-2} = 36[/latex] [latex]x^{-2} = (frac{1}{6})^{-2}[/latex] [latex]x = frac{1}{6}[/latex]
