[tex]zad.a\\\\4log_{2} x=log_{2} 81~~zal.~~x>0\\\\log_{2} x^{4} =log_{2} 3^{4} ~~\Leftrightarrow~~x^{4} =3^{4} ~~\Rightarrow ~~x=3\\\\x=3~~\land~~x>0 ~~\Rightarrow~~ x=3\\\\Odp:~~x=3\\\\zad.b\\\\log_{4} (x+4)-log_{4}(x-1)=2~~zal.~~x+4>0,~~x-1>0~~\Rightarrow~~x>1\\\\log_{4}\frac{x+4}{x-1} =2 \\z de.logarytmow\\\\\dfrac{x+4}{x-1}=4^{2} \\\\\dfrac{x+4}{x-1}=16\\\\x+4=16(x-1)\\\\x+4=16x-16\\\\x-16x=-16-4\\\\-15x=-20~~\vert \div (-15)\\\\x=\dfrac{4}{3} \\\\[/tex]
[tex]x=1\dfrac{1}{3}~~\land ~~x>1~~\Rightarrow ~~x=1\dfrac{1}{3}\\\\Odp:~~x=1\dfrac{1}{3}\\\\zad.c\\\\log_{2} (x^{2} -6)=3+log_{2} (x-1)\\\\zal.~~x^{2} -6>0,~~x-1>0\\\\~~~~(x-\sqrt{6} )(x+\sqrt{6} )>0,~~x>1~~\Rightarrow ~~x>\sqrt{6} \\\\log_{2} (x^{2} -6)=3\cdot log_{2}2 +log_{2} (x-1)\\\\log_{2} (x^{2} -6)=log_{2} 2^{3} +log_{2} (x-1)\\\\log_{2} (x^{2} -6)=log_{2}8 +log_{2} (x-1)\\\\log_{2} (x^{2} -6)=log_{2} [8\cdot (x-1)]\\\\[/tex]
[tex]log_{2} (x^{2} -6)=log_{2} (8x-8)~~\Leftrightarrow ~~ x^{2} -6=8x-8\\\\x^{2} -6-8x+8=0\\\\x^{2} -8x+2=0\\\\\Delta=(-8)^{2} -4\cdot 1\cdot 2=64-8=56\\\\\sqrt{\Delta} =\sqrt{56} =2\sqrt{14} \\\\x_{1} =\dfrac{8-2\sqrt{14} }{2} ~~\lor~~ x_{2} =\dfrac{8+2\sqrt{14} }{2} \\\\(~~x_{1} =4-\sqrt{14} ~~\lor~~ x_{3} =4+\sqrt{14} ~~)~~\land ~~x>\sqrt{6} ~~\Rightarrow ~~x=4+\sqrt{14}\\\\Odp:~~x=4+\sqrt{14}[/tex]
korzystam ze wzorów:
[tex]log_{a} a=1\\\\log_{a} b^{c} =c\cdot log_{a} b\\\\log_{a} b-log_{a} c=log_{a} (b\div c)=log_{a}\dfrac{b}{c} \\\\log_{a} b+log_{a} c=log_{a} (b\cdot c)[/tex]
