Có bao nhiêu số nguyên x thoả mãn điều kiện 7^x - 49log3^2x — Không quảng cáo

Phương pháp giải

Giải bất phương trình \(A.B < 0 \Leftrightarrow \left[ {\begin{array}{*{20}{l}}{\left\{ {\begin{array}{*{20}{l}}{A > 0}\\{B < 0}\end{array}} \right.}\\{\left\{ {\begin{array}{*{20}{l}}{A < 0}\\{B > 0}\end{array}} \right.}\end{array}} \right.\).

Điều kiện: \(x > 0\)

\(\left( {{7^x} - 49} \right)\left( {{\rm{log}}_3^2x - 7{\rm{lo}}{{\rm{g}}_3}x + 6} \right) < 0 \Leftrightarrow \left[ {\begin{array}{*{20}{c}}{\left\{ {\begin{array}{*{20}{c}}{{7^x} - 49 > 0}\\{{\rm{log}}_3^2x - 7{\rm{lo}}{{\rm{g}}_3}x + 6 < 0}\end{array}} \right.}\\{\left\{ {\begin{array}{*{20}{c}}{{7^x} - 49 < 0}\\{{\rm{log}}_3^2x - 7{\rm{lo}}{{\rm{g}}_3}x + 6 > 0}\end{array}} \right.}\end{array}} \right.\)

\(\left[ {\begin{array}{*{20}{c}}{\left\{ {\begin{array}{*{20}{c}}{{7^x} > 49}\\{1 < {\rm{lo}}{{\rm{g}}_3}x < 6}\end{array}} \right.}\\{ \Leftrightarrow \left[ {\begin{array}{*{20}{c}}{{7^x} < 49}\\{\left\{ {\begin{array}{*{20}{c}}{x > 2}\\{3 < x < {3^6}}\end{array}} \right.}\end{array}} \right.}\end{array} \Leftrightarrow \left[ {\begin{array}{*{20}{c}}{{\rm{lo}}{{\rm{g}}_3}x < 1}\\{{\rm{lo}}{{\rm{g}}_3}x > 6}\end{array}} \right]\left\{ {\begin{array}{*{20}{c}}{x < 2}\\{\left[ {\begin{array}{*{20}{c}}{0 < x < 3}\\{x > {3^6}}\end{array}} \right.}\end{array}} \right.} \right.\)

\( \Leftrightarrow \left[ {\begin{array}{*{20}{c}}{0 < x < 2}\\{3 < x < {3^6}}\end{array}} \right.\)

Mà \(x \in \mathbb{Z} \Rightarrow x \in \left\{ {1;4;5; \ldots ;728} \right\}\)

Vậy có 726 số thỏa mãn.

Đáp án B.