[tex]zad.20\\\\najpierw~~oblicze~~wartosc~~wyrazenia:\\\\\dfrac{log_{3}25\cdot log_{3} 125 }{(log_{3} 0,5+log_{3} 10)^{2} } =\dfrac{log_{3}5^{2} \cdot log_{3} 5^{3} }{(log_{3} (\frac{1}{2} \cdot 10) )^{2} } =\\\\\\=\dfrac{2log_{3}5 \cdot 3log_{3} 5 }{(log_{3} 5 )^{2} } =\dfrac{6\cdot (log_{3} 5 )^{2}}{(log_{3} 5 )^{2}} =6\\\\\\log_{36} \dfrac{log_{3}25\cdot log_{3} 125 }{(log_{3} 0,5+log_{3} 10)^{2} } =log_{36} 6=\dfrac{1}{2} \\\\\\[/tex]
[tex]log_{36}6 =x~\Rightarrow ~~36^{x} =6~~\Rightarrow ~~6^{2x} =6 ~~\Leftrightarrow ~~2x=1 ~\Rightarrow ~~x=\dfrac{1}{2}\\\\lub\\\\log_{36}6 =log_{36}\sqrt{36} =log_{36}36^{\frac{1}{2} } =\dfrac{1}{2} \cdot log_{36}36=\dfrac{1}{2} \cdot 1 = \dfrac{1}{2}[/tex]
korzystam ze wzorów:
[tex]log_{a} b=c~~\Leftrightarrow ~~a^{c} =b~~zal.~~a>0,~~a\neq 1,~~b>0\\\\log_{a} b +log_{a} c=log_{a} (b\cdot c)\\\\log_{a} a=1\\\\log_{a} b^{c} =c\cdot log_{a}b[/tex]
zad.21
[tex]\dfrac{9^{-\frac{1}{3} }\cdot \sqrt[6]{3} }{\sqrt{3^{-9} } \cdot 9^{\frac{2}{3} }\cdot \sqrt[3]{27^{2} } } =\dfrac{(3^{2} )^{-\frac{1}{3} }\cdot 3^{\frac{1}{6} } }{(3^{-9} )^{\frac{1}{2} } \cdot (3^{2} )^{\frac{2}{3} }\cdot \sqrt[3]{(3^{3} )^{2} } } =\\\\\\=\dfrac{3^{-\frac{2}{3} } \cdot 3^{\frac{1}{6} } }{3^{-\frac{9}{2} } \cdot 3^{\frac{4}{3} } \cdot \sqrt[3]{3^{6} } }} =\dfrac{3^{-\frac{2}{3} } \cdot 3^{\frac{1}{6} } }{3^{-\frac{9}{2} } \cdot 3^{\frac{4}{3} } \cdot 3^{2} }} =\\\\\\[/tex]
[tex]=\dfrac{3^{-\frac{2}{3} +\frac{1}{6} } }{3^{-\frac{9}{2} +\frac{4}{3} +2} } =\dfrac{3^{-\frac{4}{6} +\frac{1}{6} } }{3^{-4\frac{1}{2} +1\frac{1}{3} +2} }=\dfrac{3^{-\frac{3}{6} } }{3^{-4\frac{3}{6} +1\frac{2}{6} +2}} =\dfrac{3^{-\frac{3}{6} } }{3^{-1\frac{1}{6} } } =\\\\\\=3^{-\frac{3}{6} }\div 3^{-1\frac{1}{6} }= 3^{-\frac{3}{6} -(-1\frac{1}{6} )} =3^{-\frac{3}{6} +1\frac{1}{6} } =3^{-\frac{3}{6} +\frac{7}{6} } =3^{\frac{4}{6} } =3^{\frac{2}{3} } \\[/tex]
[tex]zad.a\\\\\dfrac{9^{-\frac{1}{3} }\cdot \sqrt[6]{3} }{\sqrt{3^{-9} } \cdot 9^{\frac{2}{3} }\cdot \sqrt[3]{27^{2} } } =3^{\frac{2}{3} } \\\\\\zad.b\\\\\dfrac{9^{-\frac{1}{3} }\cdot \sqrt[6]{3} }{\sqrt{3^{-9} } \cdot 9^{\frac{2}{3} }\cdot \sqrt[3]{27^{2} } } =3^{\frac{2}{3} } =3^{2\cdot \frac{1}{3} } =(3^{2} )^{\frac{1}{3} } =9^{\frac{1}{3} } =\sqrt[3]{9}[/tex]
korzystam ze wzorów:
[tex]\sqrt[n]{x} =x^{\frac{1}{n} } \\\\x^{n} \cdot x^{m} =x^{n+m} \\\\x^{n} \div x^{m}=\dfrac{x^{n} }{x^{m} } =x^{n-m}\\\\(x^{n} )^{m} =x^{n\cdot m}[/tex]
