A) At least one real solution
B) Exactly three real solutions
C) Exactly one irrational solution
D) All the above
Correct Answer: D
Solution :
For the given equation to be meaningful we must have\[x>0\]. For \[x>0\]the given equation can be written as \[\frac{3}{4}{{({{\log }_{2}}x)}^{2}}+{{\log }_{2}}x-\frac{5}{4}={{\log }_{x}}\sqrt{2}=\frac{1}{2}{{\log }_{x}}2\] Þ \[\frac{3}{4}{{t}^{2}}+t-\frac{5}{4}=\frac{1}{2}\left( \frac{1}{t} \right)\] By putting \[t={{\log }_{2}}x\] so that \[{{\log }_{x}}2=\frac{1}{t}\] because \[{{\log }_{2}}x{{\log }_{x}}2=1\]. Þ \[3{{t}^{3}}+4{{t}^{2}}-5t-2=0\,\,\,\,\Rightarrow (t-1)(t+2)(3t+1)=0\] Þ \[{{\log }_{2}}x=t=1,-2,-\frac{1}{3}\] Þ \[x=2,{{2}^{-2}},{{2}^{-1/3}}\]or \[x=2,\frac{1}{4},\frac{1}{{{2}^{1/3}}}\] Thus the given equation has exactly three real solutions out of which exactly one is irrational namely \[\frac{1}{{{2}^{1/3}}}\].
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