Có bao nhiêu cặp số nguyên $(x;y)$ thỏa mãn ${{\log }_{3}}\left(...

Đặt
Từ giả thiết

Đặt$$ $f\left( t \right)={{\log }_{3}}\left( t+1 \right)+t-8-{{\log }_{2}}\left( \dfrac{t+24}{t} \right)t>0f'\left( t \right)=\dfrac{1}{\left( t+1 \right)\ln 3}+1-\dfrac{1}{\left( \dfrac{t+24}{t} \right)\ln 2}.\left( \dfrac{-24}{{{t}^{2}}} \right)>0 ,\forall t>0\Rightarrow f\left( t \right)f\left( t \right)\le f\left( 8 \right)=0\Rightarrow 0<t\le 8y=0\Rightarrow t=x\Rightarrow x\in \left\{ 1;2;...8 \right\}(x;y)t\le 8\Rightarrow {{x}^{2}}+4{{y}^{2}}\le 8x\Rightarrow {{\left( x-4 \right)}^{2}}+4{{y}^{2}}\le 164{{y}^{2}}\le 16\Leftrightarrow -2\le y\le 2y=\pm 2\Rightarrow {{\left( x-4 \right)}^{2}}\le 0\Rightarrow x=4(x;y)y=\pm 1\Rightarrow {{\left( x-4 \right)}^{2}}\le 12\Rightarrow -2\sqrt{3}\le x-4\le 2\sqrt{3}\Rightarrow x\in \left\{ 1;2;...7 \right\}(x;y)8+2+14=24$ cặp số nguyên thỏa mãn yêu cầu bài toán.