Question

The equation $x^{({\log }_{2}x{)}^2-(11/2){\log }_{2}x+(17/2)}=8$ has

• A. at least one real solution.
• B. exactly three real solutions.
• C. exactly one irrational solutions.
• D. Exactly two real solutions.

Answer 1

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

The correct options are
A at least one real solution.
B exactly three real solutions.
C exactly one irrational solutions.

$x^{({\log }_{2}x{)}^2-(11/2){\log }_{2}x+(17/2)}=8$
$x^{({\log }_{2}x{)}^2-(11/2){\log }_{2}x+(17/2)}=2^3$
$\left\{\right({\log }_{2}x{)}^2-\left(11/2\right){\log }_{2}x+\left(17/2\right)\left\}\right({\log }_{2}x)=3$
Let ${\log }_{2}x=t$, then
$(t^2-\frac{11}{2}t+\frac{17}{2})t=3$
$2t^3-11t^2+17t-6=0$
$\Rightarrow (t-2)(t^2-\frac{7}{2}t+\frac{3}{2})=0$
$\Rightarrow (t-2)(t-\frac{1}{2})(t-3)=0$
$\Rightarrow t=\frac{1}{2},2,3$
$\Rightarrow {\log }_{2}x=\frac{1}{2},2,3$
$x=\sqrt{2},4,8$