The solution set of the inequality $\log _{10}\left(x^2-16\right) \leq \log _{10}(4 x-11)$

(4, 5]

Since base of log is same both the sides and greater than 1, hence inequality will remain same.

$\Rightarrow x^2-16 \leq 4 x-11 \Rightarrow x^2-4 x-5 \leq 0 \Rightarrow-1 \leq x \leq 5$ ……….(1)

Also $x^2 - 16 > 0$ and 4x - 11 > 0 ⇒ either x < -4 or x > 4 ……….(2)

Taking intersection of (1) and (2)

$x ∈ (4, 5]$

Hence (2) is the correct answer.