[tex]a)\\\\log_{2} 2^{2} +log_{2} 4^{3} =2\cdot log_{2} 2 +log_{2} (2^{2} )^{3}=2\cdot 1+log_{2}2^{2\cdot 3} =2+log_{2}2^{6}=2+6\cdot log_{2}2=2+6\cdot 1=2+6=8\\\\b)\\\\log_{27} 3 - log_{27} 9=log_{27} \dfrac{3}{9} =log_{27} \dfrac{1}{3} =log_{27}3^{-1} =-1\cdot log_{27}3=-1\cdot \dfrac{1}{3} =-\dfrac{1}{3} \\\\c)\\\\log_{4} 4+log_{4}\sqrt[3]{4} +log_{4}1=1+log_{4}4^{\frac{1}{3} } +0=1+\dfrac{1}{3}\cdot log_{4} 4=1+\dfrac{1}{3}\cdot 1 =1+\dfrac{1}{3}=1\dfrac{1}{3}\\\\lub[/tex]
[tex]log_{4} 4+log_{4}\sqrt[3]{4} +log_{4}1=log_{4}(4\cdot \sqrt[3]{4} )+0=log_{4}(4\cdot 4^{\frac{1}{3} } )=log_{4}4^{1+\frac{1}{3} } =log_{4}4^{1\frac{1}{3} } =1\dfrac{1}{3} \cdot log_{4} 4=1\dfrac{1}{3} \cdot 1=1\dfrac{1}{3}[/tex]
[tex]d)\\\\log_{2} 0,5^{2} -log_{2} 0,125^{3}=log_{2} (\frac{1}{2} )^{2} -log_{2} (\frac{1}{8} )^{3}=log_{2} 2^{-2} -log_{2} 8^{-3} =-2\cdot log_{2} 2 -log_{2} (2^{3} )^{-3} =-2\cdot 1 -log_{2} 2^{-9} =-2-(-9)log_{2} 2=-2-(-9)\cdot 1 = -2 +9=7\\\\e)\\\\2log_{3} 6-log_{3} 4=log_{3}6^{2} -log_{3}4=log_{3}36-log_{3}4=log_{3}\dfrac{36}{4} =log_{3}9=log_{3}3^{2} =2\cdot log_{3}3=2\cdot 1=2[/tex]
korzystałam ze wzorów:
[tex]x^{n} \cdot x^{m} =x^{n+m} \\\\log_{a} 1=0\\\\log_{a} a=1\\\\log_{a} b+log_{a} c = log_{a} (b\cdot c)\\\\log_{a} b-log_{a} c = log_{a} (b\div c)=log_{a}\dfrac{b}{c} \\\\log_{a} b^{c} =c\cdot log_{a} b\\\\\sqrt[n]{x} =x^{\frac{1}{n} } \\\\(\frac{x}{y} )^{-n} =(\frac{y}{x} )^{n}[/tex]
