Question

If $x=\log _2 3-4 \log _{\left(\ln \frac{5}{4}\right)^2} 3, y=\log _{\sqrt{3}}\left(\ln \frac{5}{4}\right)$ and $z=\log _2\left(\log _{(5 / 4)} e\right)$ then find the absolute value of $(x y+2 z)$

Answer 1

$xy +2 z =\left(\log _2 3-4 \log _{\left(\ln \frac{5}{4}\right)^3} 3\right) 2 \log _3\left(\ln \frac{5}{4}\right)+2 \log _2\left(\log _{\frac{5}{4}} e \right)$
$=2 \log _2\left(\ln \frac{5}{4}\right)-8-2 \log _2\left(\ln \frac{5}{4}\right)=-8 $
$\therefore \quad|x y+2 z|=8$