Question

Solve for ${}^{\prime }{x}^{\prime }$ if $2{\log }_{x}7+{\log }_{7x}7+3{\log }_{x}7=0$.

Answer 1

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

$\Rightarrow 5{\log }_{x}7+{\log }_{7x}7=0$

$\Rightarrow 5{\log }_{x}7+\frac{{\log }_{x}7}{{\log }_{x}7x}=0$

$\Rightarrow 5{\log }_{x}7+\frac{{\log }_{x}7}{{\log }_{x}x+{\log }_{x}7}=0$

$\Rightarrow 5{\log }_{x}7+\frac{{\log }_{x}7}{1+{\log }_{x}7}=0$ since ${\log }_{x}x=1$

Let $t={\log }_{x}7$

$\Rightarrow 5t+\frac{t}{1+t}=0$

$\Rightarrow t(5+\frac{1}{1+t})=0$

$\Rightarrow t=0,\frac{1}{1+t}=-5$

$\Rightarrow t=0,1+t=\frac{-1}{5}$

$\Rightarrow t=0,t=\frac{-1-5}{5}=\frac{-6}{5}$

$\Rightarrow {\log }_{x}7=0,{\log }_{x}7=\frac{-6}{5}$

$\Rightarrow x=7^0,x=7^{\frac{-6}{5}}$

$\therefore x=1\frac{1}{7^{\frac{6}{5}}}$