Question

The equation $\log _{ x ^2} 16+\log _{2 x } 64=3$ 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. $=\frac{1}{\log _{2^4} x^2}+\frac{1}{\log _{2^6} 2 x}=3 \Rightarrow \frac{1}{1 / 2 \log _2 x}+\frac{1}{1 / 6\left(1+\log _2 x\right)}=3$
let $\log _2 x = y$
$\Rightarrow \frac{2}{y}+\frac{6}{1+y}=3 \Rightarrow 2(1+y)+6 y=3 y(1+y) $
$\Rightarrow(y-2)(3 y+1)=0 \Rightarrow y=2 \text { or } y=-1 / 3$
$\log _2 x=2 \Rightarrow x=4 \text { and } \log _2 x=\frac{-1}{3} \Rightarrow x=2^{-1 / 3}$