Zad 1. oblicz: a)[latex]log _{5}25-log_{2}3+ frac{1}{2}log_{2}36+log_{3} sqrt{3} [/latex] b)[latex]-log _{2}(log_{2} sqrt[4]{} sqrt{2})[/latex] c)[latex]49^{log_{7} } ^{5} +10^{1-log_{5} } [/latex] d)[latex](log_{a}b+log_{b}a+2)(log_{a}b-log_{ab}b)*lo

a) [latex]log _{5}25-log_{2}3+ frac{1}{2}log_{2}36+log_{3} sqrt{3}=[/latex] [latex]log _{5}5^2-log_{2}3+ log_{2}36^{frac{1}{2}}+log_{3} 3^{frac{1}{2}}=[/latex] [latex]2log _{5}5-log_{2}3+ log_{2}6+frac{1}{2}log_{3} 3=[/latex] [latex]2-log_{2}3+ log_{2}(2 cdot 3)+frac{1}{2}=[/latex] [latex]2 frac{1}{2} -log_{2}3+ log_{2}2+log_23=[/latex] [latex]2 frac{1}{2} +1=3frac{1}{2}[/latex] b) [latex]-log _{2}(log_{2} sqrt[4]{2})=[/latex] [latex]-log _{2}(log_{2} 2^{frac{1}{4}})=[/latex] [latex]-log _{2}(frac{1}{4}log_{2} 2)=[/latex] [latex]-log _{2}frac{1}{4}=[/latex] [latex]-log _{2}2^{-2}=[/latex] [latex]-(-2)log _{2}2=2[/latex] c) [latex]49^{log_{7} } ^{5} +10^{1-log5 }=[/latex] [latex](7^2)^{log_{7} } ^{5} +10 cdot 10^{-log5}=[/latex] [latex]7^{2log_{7} } ^{5} +10 cdot 10^{log5^{-1}}=[/latex] [latex]7^{log_{7} } ^{5^2} +10 cdot 10^{log frac{1}{5} }=[/latex] [latex]7^{log_{7} } ^{25} +10 cdot frac{1}{5}=[/latex] [latex]25 +2=27[/latex] d) [latex](log_{a}b+log_{b}a+2)(log_{a}b-log_{ab}b) cdot log_{b}a-1=[/latex] [latex]left(frac{log b}{log a} + frac{log a}{log b} +2 ight) left(frac{log b}{log a} - frac{log b}{log (ab)} ight) cdot frac{log a}{log b} -1=[/latex] [latex]left(frac{log^2 b}{log a log b} + frac{log^2 a}{log a log b} + frac{2log a log b}{2log a log b} ight) cdot log b left(frac{1}{log a} - frac{1}{log (ab)} ight) cdot frac{log a}{log b} -1=[/latex] [latex]frac{log^2 b+log^2 a+ 2log a log b}{log a log b}left(frac{log(ab)}{log a log(ab)} - frac{log a}{log alog (ab)} ight) cdot log a -1=[/latex] [latex]frac{(log a+log b)^2}{log b} cdot frac{log (ab)-loga}{log alog (ab)} -1=[/latex] [latex]frac{log^2 (ab)}{log b} cdot frac{log a + log b - log a}{log a log(ab)} -1=[/latex] [latex]frac{log (ab)}{log b} cdot frac{log b}{log a}-1=[/latex] [latex]frac{log (ab)}{log a}-1=[/latex] [latex]frac{log a + log b}{log a}-1=[/latex] [latex]frac{log a}{log a}+frac{log b}{log a}-1=1+log_ab-1=log_ab[/latex] ====================== a) [latex]log_{5} 4=a Rightarrow log_{5} 2^2=a Rightarrow 2log_{5} 2=a /:2 Rightarrow log_52= frac{1}{2}a[/latex] [latex]log_{5} 3=b[/latex] [latex]log_{25}48= frac{log_548}{log_525}=frac{log_5(2^4 cdot 3)}{log_55^2}=frac{log_52^4+log_53}{2log_55}=[/latex] [latex]frac{4log_52+b}{2}=frac{2a +b}{2}[/latex] b) [latex]log_{a} x=2[/latex] [latex]log_{b} x=3[/latex] [latex]log_bx=frac{log_ax}{log_ab} Rightarrow log_ab= frac{log_ax}{log_bx}= frac{2}{3}[/latex] [latex]log_{ab} x= frac{log_ax}{log_a(ab)} =frac{2}{log_aa+log_ab} =frac{2}{1+frac{2}{3}} =frac{2}{frac{5}{3}} = frac{6}{5}[/latex] c) [latex]log_{a} x =2[/latex] [latex]log_{b} x =3[/latex] [latex]log_bx= frac{log_ax}{log_ab} Rightarrow log_ab= frac{log_ax}{log_bx}= frac{2}{3}[/latex] [latex]log_{c} x =6[/latex] [latex]log_cx= frac{log_ax}{log_ac} Rightarrow log_ac= frac{log_ax}{log_cx}= frac{2}{6}= frac{1}{3}[/latex] [latex]log_{abc} x= frac{log_ax}{log_a(abc)}= frac{2}{log_aa+log_ab+log_ac}= frac{2}{1+log_ab+log_ac}=[/latex] [latex]frac{2}{1+frac{2}{3}+frac{1}{3}}= frac{2}{2}=1[/latex] d) [latex]log_{a} x=p[/latex] [latex]log_{b} x=q[/latex] [latex]log_{abc} x=r[/latex] [latex]log_{abc}x=frac{1}{log_{x}abc}=frac{1}{log_{x}a+log_{x}b+log_{x}c}=frac{1}{frac{1}{log_{a}x}+frac{1}{log_{b}x}+frac{1}{log_{c}x}}[/latex] [latex]r= frac{1}{ frac{1}{p}+ frac{1}{q}+frac{1}{log_{c}x}}[/latex] [latex]frac{1}{p}+ frac{1}{q}+frac{1}{log_{c}x}= frac{1}{r}[/latex] [latex]frac{1}{log_{c}x}= frac{1}{r}-frac{1}{p}- frac{1}{q}[/latex] [latex]frac{1}{log_{c}x}= frac{pq}{pqr}-frac{qr}{pqr}- frac{pr}{pqr}[/latex] [latex]frac{1}{log_{c}x}= frac{pq-qr-pr}{pqr}[/latex] [latex]log_{c}x= frac{pqr}{pq-qr-pr}[/latex]

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