Question

\(\log_3\left(3^x-1\right).\log_3\left(3^{x+1}-3\right)=6\)

Answer 1

Lời giải:

Để ý rằng \(\log _3(3^{x+1}-3)=\log_3[3(3^x-1)]=1+\log_3(3^x-1)\)

Đặt \(\log_3(3^x-1)=t\). Khi đó PT tương đương:

\(t(t+1)=6\Leftrightarrow (t-2)(t+3)=0\Rightarrow \)\(\left[{}\begin{matrix}t=2\\t=-3\end{matrix}\right.\)

Nếu \(t=2\rightarrow 3^x-1=9\Leftrightarrow 3^x=10\rightarrow x=\log_3(10)\)

Nếu \(t=-3\Rightarrow 3^x-1=\frac{1}{27}\Rightarrow 3^x=\frac{28}{27}\Rightarrow x=\log_3\left (\frac{28}{27}\right)\)