Question

Cum a ajuns la forma cu paranteze din prima formă a ecuației ? Imi puteți arăta pașii vă rog ?

Answer 1

 

$\displaystyle\bf\\\frac{3}{log_{_{\bf\b5}}x}+log_{_{\bf\b5}}5x=5\\\\\frac{3}{log_{_{\bf\b5}}x}+log_{_{\bf\b5}}5x-5=0\\\\ Folosim~formula:~~log_{a}xy=log_{a}x+log_{a}y\\\\\frac{3}{log_{_{\bf\b5}}x}+log_{_{\bf\b5}}5+log_{_{\bf\b5}}x-5=0\\\\\frac{3}{log_{_{\bf\b5}}x}+1+log_{_{\bf\b5}}x-5=0\\\\\frac{3}{log_{_{\bf\b5}}x}+log_{_{\bf\b5}}x-4=0$

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$\displaystyle\bf\\Substitutie:~~~log_{_{\bf\b5}}x=t\\\\\frac{3}{t}+t-4=0~~\Big|\cdot t\\\\3+t^2-4t=0\\\\t^2-4t+3=0\\\\t^2-t-3t+3=0\\\\t(t-1)-3(t-1)=0\\\\(t-1)(t-3)=0\\\\Ne~intoarcem~la~substitutie.\\\\log_{_{\bf\b5}}x=t\\\\\boxed{\bf\Big(log_{_{\bf\b5}}x-1\Big)\Big(log_{_{\bf\b5}}x-3\Big)=0}~~~A~ajuns~la~parantezele~din~cerinta.$

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$\displaystyle\bf\\Hai~sa~continuam~rezolvarea.\\\\log_{_{\bf\b5}}x-1=0\\\\log_{_{\bf\b5}}x=1\\\\\boxed{\bf~x_1=5^1=5}\\\\log_{_{\bf\b5}}x-3=0\\\\log_{_{\bf\b5}}x=3\\\\\boxed{\bf~x_2=5^3=125}$