Question

Help urgent, am test

Answer 1

  Am extras din fisierul tau o imagine care contine doar exercitiile 2 si 3 si o gasesti atasata aici la rezolvare.

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$\displaystyle\bf\\2)~\textbf{Sa demonstram ca:}\\\\log_611=\frac{log_211\cdot log_311}{log_211+log_311}\\\\Rezolvare:\\\\\frac{log_211\cdot log_311}{log_211+log_311}=\frac{1}{~~\dfrac{log_211+ log_311}{log_211\cdot log_311}~~}=\\\\=\frac{1}{~~\dfrac{log_211}{log_211\cdot log_311}+\dfrac{log_311}{log_211\cdot log_311}~~}=\\\\\\=\frac{1}{~~\dfrac{1}{log_311}+\dfrac{1}{log_211}~~}=\frac{1}{~~log_{11}3+log_{11}2~~}=\\\\=\frac{1}{~~log_{11}(3\cdot2)~~}=\frac{1}{~~log_{11}(6)~~}=\boxed{\bf log_{6}11}$

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3)  Sa se calculeze:

$\displaystyle\bf\\log_3\frac{2}{1}-log_3\frac{2}{3}+log_3\frac{4}{3}-log_3\frac{4}{5}+log_3\frac{6}{5}-log_3\frac{6}{7}+log_3\frac{8}{7}-log_3\frac{8}{9}=\\\\=log_3\frac{\dfrac{2}{1}}{\dfrac{2}{3}}+log_3\frac{\dfrac{4}{3}}{\dfrac{4}{5}}+ log_3\frac{\dfrac{6}{5}}{\dfrac{6}{7}}+log_3\frac{\dfrac{8}{7}}{\dfrac{8}{9}}=\\\\=log_3\Big(\frac{2}{1}\times\frac{3}{2}\Big)+log_3\Big(\frac{4}{3}\times\frac{5}{4}\Big)+log_3\Big(\frac{6}{5}\times\frac{7}{6}\Big)+log_3\Big(\frac{8}{7}\times\frac{9}{8}\Big)=$

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$\displaystyle\bf\\=log_3\frac{3}{1}+log_3\frac{5}{3}+log_3\frac{7}{5}+log_3\frac{9}{7}=\\\\=log_3\left(\frac{3}{1}\times\frac{5}{3}\times\frac{7}{5}\times\frac{9}{7}\right)=\\\\Se~fac~simplificari.\\\\=log_3\frac{9}{1}=log_39=log_33^2=2log_33=2\times1=\boxed{\bf2}$