Definicja logarytmu: [latex]log_{a}{b}=c <=> a^{c}=b[/latex] Własności logarytmów: [latex]log_{a}{1}=0\ log_{a}{a}=1\ a^{log_{a}{b}}=b[/latex] Prawa działań na logarytmach: [latex]log_{a}{b_{1}*b_{2}}=log_{a}{b_{1}}+log_{a}{b_{2}}\ log_{a}{frac{b_{1}}{b_{2}}}=log_{a}{b_{1}}-log_{a}{b_{2}}\ log_{a}{b^{n}}=n*log_{a}{b}\ log_{a}{b}=frac{1}{log_{b}{a}}\ log_{a}{b}=frac{log_{c}b}{log_{c}a}[/latex] Własności działań na potęgach: [latex]a^{n}*a^{m}=a^{n+m}\ a^{n}:a^{m}=a^{n-m}\ (a^{n})^{m}=a^{n*m}\ a^{-n}=(frac{1}{a})^{n}\ a^{n}*b^{n}=(ab)^{n}[/latex] ================================================================= zad 1 log2=a i log3=b [latex]log8*log_{8}sqrt{6}=\ =log2^{3}*log_{2^{3}}6^{frac{1}{2}}=\ =3*log{2}*frac{1}{2}log_{2^{3}}6=\ =frac{3}{2}a*frac{1}{log_{6}{2^{3}}}=\ =frac{3}{2}a*frac{1}{3log_{6}{2}}=\ =frac{3}{2}a*frac{1}{3}log_{2}{6}=\ =frac{1}{2}a*log_{2}(2*3)=\ =frac{1}{2}a*(log_{2}3+log_{2}2)=\ =frac{1}{2}a*(log_{2}3+1)[/latex] --- [latex]frac{1}{2}a*(log_{2}3+1)=\ =frac{1}{2}a*(frac{log_{10}3}{log_{10}{2}}+1)=\ =frac{1}{2}a*(frac{log3}{log2}+1)=\ =frac{a}{2}*(frac{b}{a}+1)=\ =frac{a}{2}*frac{b}{a}+frac{a}{2}*1=\ =frac{b}{2}+frac{a}{2}=\ =frac{a+b}{2}[/latex] ================================= zad 2 [latex]5log_{8}2 + 2(log_{8}1-log_{8}4)=\ =log_{8}2^{5}+log_{8}frac{1}{4}=\ =log_{8}32+log_{8}frac{1}{4}=\ =log_{8}(32*frac{1}{4})=\ =log_{8}8=\ =1[/latex] ================================= zad 3 [latex]log_{2sqrt{2}}x=-3 <=> 2sqrt{2}^{-3}=x\ 2sqrt{2}^{-3}=x\ (2*2^{frac{1}{2}})^{-3}=x\ (2^{frac{3}{2}})^{-3}=x\ 2^{-frac{9}{2}}=x\ {frac{1}{2}}^{frac{9}{2}}=x\ sqrt{(frac{1}{2})^{9}}=x\ x=sqrt{frac{1}{512}}[/latex] ================================= zad 4 [latex]log^{2}2+log^{2}5+log4*log5=1\ log^{2}2+log^{2}{5}+log2^{2}*log5=1\ log^{2}2+log^{2}{5}+2log2*log5=1\ log^{2}2+2log2*log5+log^{2}5=1\ ---\ a^{2}+2ab+b^{2}=(a+b)^{2}\ ---\ (log2+log5)^{2}=1\ (log(2*5))^{2}=1\ (log10)^{2}=1\ ---\ log a=log_{10}a\ ---\ 1^{2}=1\ 1=1\ L=P [/latex]
