If x1 and x2 are the roots of the equation 2010xlog2010x=x2, then find the cyphers at the end of the product (x1x2).

Taking log on the both sides with base a, we get (where a=2010)1+logax2(logax)=2logax Let logax=y, so we get ∴y1+y2=4, so loga(x1x2)=4⇒x1x2=a4=(2010)2=4040100 Hence number of cyphers at the end of product (x1x2)=2

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If $x_{1}$ and $x_{2}$ are the roots of the equation $\sqrt{2010} x^{\log_{2010} x} = x^{2},$ then find the cyphers at the end of the product $(x_{1} x_{2}) .$

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Detailed Solution

Taking log on the both sides with base a, we get (where $a = \sqrt{2010}$ )
$1 + \frac{\log_{a} x}{2} (\log_{a} x) = 2 \log_{a} x$
Let $\log_{a} x = y,$ so we get Question Image
$∴ y_{1} + y_{2} = 4,$ so
$\log_{a} (x_{1} x_{2}) = 4 \Rightarrow x_{1} x_{2} = a^{4} = {(2010)}^{2} = 4040100$
Hence number of cyphers at the end of product $(x_{1} x_{2}) = 2$