A) at least one real solutions
B) exactly three real solutions
C) exactly one irrational solution
D) complex roots
Correct Answer: B
Solution :
\[\frac{3}{4}{{({{\log }_{2}}x)}^{2}}+{{\log }_{2}}x-\frac{5}{4}={{\log }_{x}}\sqrt{2}\] \[\Rightarrow \]\[\frac{3}{4}{{({{\log }_{2}}x)}^{2}}+{{\log }_{2}}x-\frac{5}{4}=\frac{1}{2{{\log }_{2}}x}\] \[\Rightarrow \]\[3{{({{\log }_{2}}x)}^{3}}+4{{({{\log }_{2}}x)}^{2}}-5({{\log }_{2}}x)-2=0\] Put \[lo{{g}_{2}}x=y\] \[\therefore \] \[3{{y}^{3}}+4{{y}^{2}}-5y-2=0\] \[\Rightarrow \] \[(y-1)(y+2)(3y+1)=0\] \[\Rightarrow \] \[y=1,-2,\frac{-1}{3}\] \[\Rightarrow \] \[{{\log }_{2}}x=1,-2,-\frac{1}{3}\] \[\Rightarrow \] \[x=2,\frac{1}{{{2}^{1/3}}},\frac{1}{4}\]
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