Question

The equation ${\log }_{x^2}16+{\log }_{2x}64=3$ has

• A. one irrational solution
• B. no prime solution
• C. two real solutions
• D. no integral solution

Answer 1

Answer: (A), (B), (C)

L.H.S. $=\frac{1}{{\log }_{2^4}{x}^2}+\frac{1}{{\log }_{2^6}2x}=3\Rightarrow \frac{1}{1/2{\log }_{2}x}+\frac{1}{1/6\left(1+{\log }_{2}x\right)}=3$
let ${\log }_{2}x=y$
$\Rightarrow \frac{2}{y}+\frac{6}{1+y}=3\Rightarrow 2(1+y)+6y=3y(1+y)$
$\Rightarrow (y-2)(3y+1)=0\Rightarrow y=2\text{ or }y=-1/3$
${\log }_{2}x=2\Rightarrow x=4\text{ and }{\log }_{2}x=\frac{-1}{3}\Rightarrow x=2^{-1/3}$