If (√log x to the base 2) - 0.5 = log √x to the base 2 , then x equals

(1) odd integer

(2) prime number

(3) composite number

(4) irrational

We are given the equation:

√(log₂ x) - 0.5 = log₂(√x)

Step 1: Introduce a substitution
Let y = log₂ x. This means:

x = 2ʸ
log₂(√x) = log₂(2^(y/2)) = y/2
Thus, rewriting the given equation using y:

√y - 0.5 = y/2

Step 2: Solve for y
Rearrange the equation:

√y = y/2 + 0.5

Square both sides to eliminate the square root:

(√y)² = (y/2 + 0.5)²

y = (y²/4) + (y/2) + 0.25

Multiply everything by 4 to remove fractions:

4y = y² + 2y + 1

Rearrange into standard quadratic form:

y² - 2y + 1 = 0

(y - 1)² = 0

So, y = 1.

Step 3: Solve for x
Since y = log₂ x, we substitute:

log₂ x = 1

This means:

x = 2¹ = 2

Step 4: Identify the type of number
The number 2 is a prime number.

Final Answer:
Option (2) prime number.