Question

The number of real values of the parameter k for which (log x at base 16)^2 - log x at base 16+ log k at base 16 =0 with real coefficients will have exactly one solution is

Answer 1

(Log₁₆ x )² - (Log₁₆ x)  + Log₁₆ K  = 0

Clearly  x and K are positive real numbers.

Let  Log₁₆ x = y    then  :   y² -y + Log₁₆ K =0

Discriminant :  1 - 4 * Log₁₆ K  = 0  for the quadratic to have exactly one solution.

so  Log₁₆ K = 1/4    =>   K  = 16¹/⁴  = 2

In that case,   y = 1/2  and  x = 4