Wiedząc, że: a) log₁₄2 = a , log₁₄5 = b ; oblicz log₇50 b) log₃20 = a , log₃15 = b ; oblicz log₂360

a) [latex]log_{14}2 = a[/latex] [latex]log_{14}5 = b[/latex] [latex]log_750= frac{log_{14} 50}{log_{14}7}=frac{log_{14}(2cdot 25)}{log_{14} frac{14}{2} }=[/latex] [latex]frac{log_{14}(2cdot 5^2)}{log_{14}14-log_{14}2}=frac{log_{14}2 +log_{14} 5^2}{1-a}=[/latex] [latex]frac{a+2log_{14} 5}{1-a}=frac{a+2b}{1-a}=[/latex] b) [latex]log_320 = a[/latex] [latex]log_315 = b[/latex] [latex]log_2360=frac{log_3360}{log_32}=frac{log_3(20cdot 18)}{log_32}=[/latex] [latex]frac{a+log_318}{log_32}=frac{log_320+log_3(2cdot 9)}{log_32}=[/latex] [latex]frac{a+log_32+log_39}{log_32}=frac{a+log_32+log_33^2}{log_32}=[/latex] [latex]frac{a+log_32+2log_33}{log_32}=frac{a+log_32+2}{log_32}=[/latex] ====================== [latex]log_315=b[/latex] [latex]log_3(3cdot 5)=b[/latex] [latex]log_33+log_35=b[/latex] [latex]1+log_35=b[/latex] [latex]log_35=b-1[/latex] -------------------------- [latex]log_320=a[/latex] [latex]log_3(4cdot 5)=a[/latex] [latex]log_34+log_35=a[/latex] [latex]log_32^2+log_35=a[/latex] [latex]2log_32+log_35=a[/latex] [latex]2log_32=a-log_35[/latex] [latex]2log_32=a-(b-1)[/latex] [latex]2log_32=a-b+1 /:2[/latex] [latex]log_32= frac{a-b+1}{2}[/latex] ====================== [latex]frac{a+log_32+2}{log_32}= frac{a+frac{a-b+1}{2}+2}{frac{a-b+1}{2}}=[/latex] [latex]frac{2a+a-b+1+4}{a-b+1}=frac{3a-b+5}{a-b+1}[/latex]