Log question?

Algebra Level 4

Let $p$ denote the product of roots to the equation $\large x^{\log_{10} x} = 2010$ . Find the value of $10p^2$ .

1 solution

Kwg Dennis
Dec 27, 2015

Take log from both sides.

$logx^{logx}=log2010$

$logx \times logx=log2010$

$(logx)^2=log2010$

$logx=\sqrt{log2010}$ or $logx= -\sqrt{log2010}$

x= $10^{\sqrt{log2010}}$ or $10^{-\sqrt{log2010}}$

Product of roots: p= $10^{\sqrt{log2010}+(-\sqrt{log2010})}$ =1

$10p^2 = 10 \times 1=10$