Question

The number of real values of the parameter $k(k>0)$ for which $({\log }_{16}x{)}^2-{\log }_{16}x+{\log }_{16}k=0$ has exactly one solution for $x$ is

• A. $2$
• B. $1$
• C. $4$
• D. none of these

Answer 1

Answer: (B)

$1$

Let ${\log }_{16}x=t$
Hence,
$t^2-t+{\log }_{16}k=0$
$k>0$
The equation has exactly one solution. Hence, the discriminant must be zero.
$\Delta =1-4{\log }_{16}k$
$\Rightarrow 0=1-4{\log }_{16}k$
$\Rightarrow {\log }_{16}k=\frac{1}{4}$
$\Rightarrow k=2$
Hence, option B is correct