2.345 [latex]f) log_{sqrt{3}}x=5 <=> sqrt{3}^{5}=x\ sqrt{3}^{5}=x\ x=3^{2}*sqrt{3}\ x=9sqrt{3}\ ----------\ i) log_{9}(log_{4}x)=-frac{1}{2}\ log_{4}x=9^{-frac{1}{2}}\ log_{4}x=sqrt{frac{1}{9}}\ log_{4}x=frac{1}{3} <=> 4^{frac{1}{3}}=x\ 4^{frac{1}{3}}=x\ x=sqrt[3]{4}[/latex] [latex]g) log_{frac{3}{16}}x=frac{3}{2} <=> frac{3}{16}^{frac{3}{2}}=x\ frac{3}{16}^{frac{3}{2}}=x\ x=sqrt{(frac{3}{16})^{3}}\ x=sqrt{frac{27}{4096}}\ x=frac{3sqrt{3}}{64}[/latex] ================================= 2.346 [latex]a) log_{x}16=4 <=> x^{4}=16\ x^{4}=16\ x^{4}=2^{4}\ x=2\ ----------\ b) log_{x}0,09=frac{2}{3} <=>x^{frac{2}{3}}=frac{9}{100}\ x^{frac{2}{3}}=frac{9}{100}\ x^{frac{2}{3}}=(frac{3}{10})^{2}\ x^{frac{2}{3}}=(((frac{3}{10})^{3})^{2})^{frac{1}{3}}\ x^{frac{2}{3}}=(frac{27}{1000})^{frac{2}{3}}\ x=frac{27}{1000}\ ----------\[/latex] [latex]c) log_{x}frac{1}{16}=-frac{4}{5} <=> x^{-frac{4}{5}}=frac{1}{16}\ x^{-frac{4}{5}}=frac{1}{16}\ x^{-frac{4}{5}}=4^{-2}\ x^{-frac{4}{5}}=((4^{frac{5}{2}})^{-2})^{frac{2}{5}}\ x^{-frac{4}{5}}=(2^{2*frac{5}{2}})^{-frac{4}{5}}\ x^{-frac{4}{5}}=(2^{5})^{-frac{4}{5}}\ x=2^{5}\ x=32\ ----------\ [/latex] [latex]d) log_{x}sqrt{5}=frac{1}{4} <=> x^{frac{1}{4}}=5^{frac{1}{2}}\ x^{frac{1}{4}}=5^{frac{1}{2}}\ x^{frac{1}{4}}=((5^{2})^{frac{1}{2}})^{frac{1}{2}}\ x^{frac{1}{4}}=(5^{2})^{frac{1}{4}}\ x=5^{2}\ x=25\----------\[/latex] [latex]e) log_{x}2sqrt{8}=5 <=> x^{5}=2*2^{frac{3}{2}}\ x^{5}=2^{1+frac{3}{2}}\ x^{5}=2^{frac{5}{2}}\ x^{5}=sqrt{2}^{5}\ x=sqrt{2}\ ----------\[/latex] [latex]f) log_{x}49=4 <=> x^{4}=7^{2}\ x^{4}=((7^{2})^{2})^{frac{1}{2}}\ x^{4}=sqrt{7}^{4}\ x=sqrt{7}\ ----------\ g) log_{x}frac{1}{12}=-2 <=> x^{-2}=12^{-1}\ x^{-2}=((12^{-1})^{2})^{frac{1}{2}}\ x^{-2}=sqrt{12}^{-2}\ x=sqrt{12}\ ----------\[/latex] [latex]h) log_{a}a=1\ log_{x-1}10=1 => x-1=10 => x=11\ ----------\ i) log_{sqrt{2}}(log_{x}9)=2\ log_{x}9=sqrt{2}^{2}\ log_{x}9=2 <=> x^{2}=9\ x^{2}=3^{2}\ x=3[/latex]
