Question

Find the rational value of x if 6(${\log }_x$2-${\log }_4$ x)+7=0

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Answer 1

6(${\log }_x$2-${\log }_4$ x)+7=0

6${\log }_{x}2$-6${\log }_{4}x$ + 7=0

6${\log }_{x}2$-6${\log }_{2^2}x$ + 7=0

6${\log }_{x}2$-$\frac{6}{2}$${\log }_{2}x$ + 7=0 $\{{\log }_{a^k}x=\frac{1}{k}{\log }_{a}x\}$

$\frac{6}{{\log }_{2}x}$ - 3${\log }_{2}x$ + 7 = 0

Let ${\log }_{2}x$ = y

$\frac{6}{y}$ - 3y + 7 = 0
$\Rightarrow 3y^2-7y-6=0\Rightarrow 3y^2-9y+2y-6=0\Rightarrow 3y(y-3)+2(y-3)=0\Rightarrow (y-3)(3y+2)=0\Rightarrow y=3,\frac{-2}{3}wheny=3,lo{g}_{2}x=3\Rightarrow x=2^3=8wheny=\frac{-2}{3},lo{g}_{2}x=\frac{-2}{3}x=2^{-2/3}$