Rozwiązania w załacznikach
zadanie 1 Korzystamy z definicji logarytmu: [latex]log_{a}b=c iff a^{c}=b [/latex] a) [latex]log_{frac{1}{2}}32=x\\(frac{1}{2})^{x}=32\\(2^{-1})^{x}=32\2^{-x}=32\2^{-x}=2^{5}\-x=5 /:(-1)\x=-5 o log_{frac{1}{2}}32=-5 [/latex] b) Jeśli nie została podana podstawa logarytmu to jest ona równa 10: [latex] log 1000=x\log_{10}1000=x\10^{x}=1000\10^{x}=10^{3}\x=3 o log1000=3[/latex] c) [latex]log_{frac{2}{3}} frac{81}{16}=x\\(frac{2}{3})^{x}=frac{81}{16}\\(frac{2}{3})^{x}=(frac{3}{2})^{4}\\(frac{2}{3})^{x}=((frac{2}{3}))^{-1})^{4}\\(frac{2}{3})^{x}=(frac{2}{3})^{-4}\x=-4 o log_{frac{2}{3}}frac{81}{16}=-4 [/latex] d) [latex]log2 frac{1}{1024}=x\\2^{x}=frac{1}{1024}\\((frac{1}{2})^{-1})^{x}=(frac{1}{2})^{10}\\(frac{1}{2})^{-x}=(frac{1}{2})^{10}\\-x=10 /:(-1)\x=-10 o log2 frac{1}{1024}=-10 [/latex] e) [latex]log_{frac{1}{6}} 216=x\\(frac{1}{6})^{x}=216\\(frac{1}{6})^{x}=216\\(6^{-1})^{x}=216\6^{-x}=216\6^{-x}=6^{3}\-x=3 /:(-1)\x=-3 o log_{frac{1}{6}}216=-3[/latex] f) [latex]log_{5}625=x\5^{x}=625\5^{x}=5^{4}\x=4 o log_{5}625=4[/latex] g) [latex]log_{frac{1}{5}}1=x\\(frac{1}{5})^{x}=1\x=0 o log_{frac{1}{5}}1=0[/latex] zadanie 2 Korzystamy z definicji logarytmu: [latex]log_{a}b=c iff a^{c}=b [/latex] a) [latex] log_{5} x = 3 \x=5^{3}\x=125 o log_{5} 125 = 3[/latex] b) [latex]log_{2} x = - frac{2}{3}\\x=2^{-frac{2}{3}}\\x=(frac{1}{2})^{frac{2}{3}}\\x=sqrt[3]{(frac{1}{2})^{2}}\\x=sqrt[3]{frac{1}{4}} o log_{2} sqrt[3]{frac{1}{4}} = - frac{2}{3} [/latex] c) [latex]log_{3}x= 10\x=3^{10}\x=59049 o log_{3}59049= 10 [/latex] d) [latex]log_{2sqrt{2}} x = -3\x=(2sqrt{2})^{-3}\x=(frac{1}{2sqrt{2}})^{3}\\x=frac{1}{8*2sqrt{2}}\\x=frac{1}{16sqrt{2}}\\x=frac{sqrt{2}}{16sqrt{2}*sqrt{2}} \\x=frac{sqrt{2}}{32}[/latex] zadanie 3 Korzystamy z definicji logarytmu: [latex]log_{a}b=c iff a^{c}=b [/latex] a) [latex] log_{x} 25 = 2\\x^{2}=25\x=5 lub x=-5 <0\x=5 o log_{5}25=2[/latex] b) [latex] log_{x} frac{1}{9} = -2\\x^{-2}=frac{1}{9}\\(frac{1}{x})^{2}=(frac{1}{3})^{2}\\x=3 o log_{3}frac{1}{9}=-2[/latex] c) [latex]log_{x} 27= 3 \x^{3}=27\x^{3}=3^{3}\x=3 o log_{3}27=3[/latex] d) [latex] log_{x} 36 = -2\x^{-2}=36\x^{-2}=6^{2}\x^{-2}=(frac{1}{6})^{-2}\\x=frac{1}{6} o log_{frac{1}{6}}36=-2[/latex]
