Có bao nhiêu số nguyên x thỏa mãn ${\text{log}}_3\frac{x^2-16}{343}<{\text{log}}_7\frac{x^2-16}{27}$?

Chọn D

TXĐ: $D=\left(-\infty ;-4\right)\cup \left(4;+\infty \right).$

Ta có:

$\begin{array}{l}{\text{log}}_3\frac{x^2-16}{343}<{\text{log}}_7\frac{x^2-16}{27}\\ \iff {\text{log}}_{3}7.\left[{\text{log}}_7\left(x^2-16\right)-3\right]<{\text{log}}_7\left(x^2-16\right)-3{\text{log}}_{7}3\\ \iff \left({\text{log}}_{3}7-1\right){\text{.log}}_7\left(x^2-16\right)<3{\text{log}}_{3}7-3{\text{log}}_{7}3\\ \iff {\log }_7\left(x^2-16\right)<\frac{3\left({\log }_{3}7-{\log }_{7}3\right)}{{\log }_{3}7-1}\\ \iff {\log }_7\left(x^2-16\right)<3\left(1+{\log }_{7}3\right)\\ \iff {\log }_7\left(x^2-16\right)<{\log }_7{21}^3\\ \iff x^2-16<{21}^3\\ \iff -\sqrt{9277}<x<\sqrt{9277}\end{array}$

Kết hợp điều kiện ta có $x\in \left\{-96;-95;\mathrm{...};-5;5;\mathrm{...};95;96\right\}$. Vậy có 184 số nguyên x thỏa mãn.