Given that \(\log_{3} x - 3\log_{x} 3 + 2 = 0\), find the values of x.
Question 1 Report
Given that \(\log_{3} x - 3\log_{x} 3 + 2 = 0\), find the values of x.
Answer Details
Note that \(\log_{x}3=\dfrac{1}{\log_{3}x}\). Let \(y=\log_{3}x\). The equation becomes
\[y-\frac{3}{y}+2=0\]
Multiply through by \(y\):
\[y^{2}+2y-3=0\Rightarrow(y+3)(y-1)=0\Rightarrow y=-3\ \text{or}\ y=1\]
Convert back with \(y=\log_{3}x\):
\[y=1\Rightarrow x=3^{1}=3,\qquad y=-3\Rightarrow x=3^{-3}=\frac{1}{27}\]
\[x=3\quad\text{or}\quad x=\frac{1}{27}\]
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