At 298 K, a certain buffer solution contains equal concentrations...

Given $K_b$ for $X^{-} = 10^{-10}$.

For a conjugate acid-base pair at $298$ K, $K_a \times K_b = K_w = 10^{-14}$.

$K_a = \dfrac{10^{-14}}{10^{-10}} = 10^{-4}$

The $pK_a$ of the weak acid $HX$ is:

$pK_a = -\log(K_a) = -\log(10^{-4}) = 4$

Using the Henderson-Hasselbalch equation for an acidic buffer:

$\text{pH} = pK_a + \log \left( \dfrac{[X^{-}]}{[HX]} \right)$

Since the concentrations of $X^{-}$ and $HX$ are equal, $[X^{-}] = [HX]$.

$\text{pH} = 4 + \log(1) = 4 + 0 = 4$

Answer: $4$