Oblicz logarytm w załączniku. Prosze o rozpisanie.

Korzystamy z definicji logarytmu: [latex]log_{a}b=c iff a^{c}=b[/latex] Jeżeli nie została podana podstawa logarytmu to wynosi ona 10 1 rząd [latex]log_{frac{1}{3}}3=x\(frac{1}{3})^{x}=3\(3^{-1})^{x}=3\3^{-x}=3^{1}\-x=1 /:(-1)\x=-1 o underline{underline{log_{frac{1}{3}}3=-1}}\\log_{0,2}125=x\0,2^{x}=125\(frac{1}{5})^{x}=5^{3}\(5^{-1})^{x}=5^{3}\5^{-x}=5^{3}\-x=3 /:(-1)\x=-3 o underline{underline{log_{0,2}125=-3}}[/latex] [latex]log_{frac{1}{3}}sqrt{3}=x\(frac{1}{3})^{x}=sqrt{3}\(3^{-1})^{x}=3^{frac{1}{2}}\3^{-x}=3^{frac{1}{2}}\-x=frac{1}{2} /:(-1)\x=-frac{1}{2} o underline{underline{log_{frac{1}{3}}sqrt{3}=-frac{1}{2}}}[/latex] 2 rząd [latex]log_{0,1}10^{6}=x\0,1^{x}=10^{6}\(frac{1}{10})^{x}=10^{6}\(10^{-1})^{x}=10^{6}\10^{-x}=10^{6}\-x=6 /:(-1)\x=-6 o underline{underline{log_{0,1}10^{6}=-6}}\\log_{frac{1}{6}}frac{1}{216}=x\(frac{1}{6})^{x}=frac{1}{216}\(frac{1}{6})^{x}=(frac{1}{6})^{3}\x=3 o underline{underline{log_{frac{1}{6}}frac{1}{216}=3}}[/latex] [latex]log_{sqrt{2}}32=x\(sqrt{2})^{x}=32\(2^{frac{1}{2}})^{x}=2^{5}\2^{frac{1}{2}x}=2^{5}\frac{1}{2}x=5 /*2\x=10 o underline{underline{log_{sqrt{2}}32=10}}[/latex] 3 rząd [latex]log_{0,5}frac{1}{128}=x\0,5^{x}=frac{1}{128}\(frac{1}{2})^{x}=(frac{1}{2})^{7}\x=7 o underline{underline{log_{0,5}frac{1}{128}=7}}\\log_{3}sqrt{3}=x\3^{x}=sqrt{3}\3^{x}=3^{frac{1}{2}} \x=frac{1}{2} o underline{underline{log_{3}sqrt{3}=frac{1}{2}}}[/latex] [latex]log1000=x\log_{10}1000=x\10^{x}=1000\10^{x}=10^{3}\x=3 o underline{underline{log1000=3}}[/latex] 4 rząd [latex]log_{0,25}16=x\0,25^{x}=16\(frac{1}{4})^{x}=4^{2}\(4^{-1})^{x}=4^{2}\4^{-x}=4^{2}\-x=2 /:(-1)\x=-2 o underline{underline{log_{0,25}16=-2}}\\log_{frac{1}{2}}2sqrt{2}=x\(frac{1}{2})^{x}=2sqrt{2}\(2^{-1})^{x}=2^{1}*2^{frac{1}{2}}\2^{-x}=2^{frac{3}{2}}\-x=frac{3}{2} /:(-1)\x=-frac{3}{2} o underline{underline{log_{frac{1}{2}}2sqrt{2}=-frac{3}{2}}}[/latex] [latex]logsqrt{10}=x\log_{10}sqrt{10}=x\10^{x}=sqrt{10}\10^{x}=10^{frac{1}{2}}\x=frac{1}{2} o underline{underline{logsqrt{10}=frac{1}{2}}}[/latex]