Zadania w załączniku (logarytmy, potęgi) proszę o pomoc :))

10. a) [latex]frac2{sqrt[3]{32}}=frac2{2^{frac53}}=2^{1-frac53}=2^{-frac23} [/latex] b) [latex]sqrt{frac12cdotsqrt{frac12}}=sqrt{2^{-1}cdotsqrt{2^{-1}}}=sqrt{2^{-1}cdot2^{-frac12}}=sqrt{2^{-1-frac12}}=sqrt{2^{-frac32}}=\=(2^{-frac32})^{frac12}=2^{-frac34[/latex] c) [latex]frac{4cdot32cdotfrac1{sqrt2}}{(2^3)^4:sqrt[3]{64}}=frac{2^2cdot2^5cdot2^{-frac12}}{2^{12}:4}=frac{2^{2+5-frac12}}{2^{12}:2^2}=frac{2^{frac{13}2}}{2^{10}}=2^{frac{13}2-10}=2^{-frac72}[/latex] 11. a) [latex]7^{2-sqrt3}cdot49^{1+frac{sqrt3}2}=7^{2-sqrt3}cdot(7^2)^{1+frac{sqrt3}2}=7^{2-sqrt3}cdot7^{2+sqrt3}=7^{2-sqrt3+2+sqrt3}=\=7^4=2401[/latex] b) [latex]log_a216=3 o a^3=216 o a^3=6^3 o a=6[/latex] c) [latex]log_{sqrt2}x=6 o x=(sqrt2)^6 o x=(2^{frac12})^6 o x=2^3=8[/latex] d) [latex]log_asqrt3=frac14 o a^{frac14}=sqrt3 o (a^{frac14})^4=(sqrt3)^4 o a=(3^{frac12})^4=3^2=9[/latex] e) [latex]log_42-log_432-6underbrace{log_41}_0=log_4(frac2{32})-6cdot0=log_4(frac1{16})-0=\=log_4(4^{-2})-0=-2-0=-2[/latex] f) [latex]log_312+2log_36-4log_32=log_312+log_36^2-log_32^4=\=log_312+log_336-log_316=log_3(frac{12cdot36}{16})=log_327=3[/latex] 12. a) [latex]log_3x=2+log_32 \ log_3x-log_32=2 \ log_3(frac x2)=2 o frac x2=3^2 o frac x2=9 o x=18[/latex] b) [latex]log x=-1+2log5 \ log x-2log5=-1 \ log x-log5^2=-1 \ log x-log25=-1 \ log (frac x{25})=-1 o frac x{25}=10^{-1} o x=25cdot10^{-1}=25cdotfrac1{10}=2.5[/latex] 13. [latex]overline{x}=frac{log_412-log_43+2}3=frac{log_4(frac{12}3)+2}3=frac{log_44+2}3=frac{1+2}3=frac33=1[/latex] 14. [latex]frac{log3+log4}{log24-log2}=frac{log(3cdot4)}{log(frac{24}2)}=frac{log12}{log12}=1[/latex] 15. [latex]a=|underbrace{sqrt5-3}_{ujemna}|+|underbrace{sqrt5-2}_{dodatnia}|=-(sqrt5-3)+(sqrt5-2)=\=-sqrt5+3+sqrt5-2=1[/latex] 16. [latex]sqrt{(x-1)^2}=sqrt{1frac9{16}} \ |x-1|=sqrt{frac{25}{16}} \ |x-1|=frac54 \ x-1=frac54 vee x-1=-frac54 \ x=1+frac54 vee x=1-frac54 \ x=frac94 vee x=-frac14[/latex]