Given that \(\log_{3} x – 3\log_{x} 3 + 2 = 0\), find the values of x.
Explanation
\(\log_{3} x - 3\log_{x} 3 + 2 = 0\)
\(\log_{3} x - 3\log_{x} 3 + 2 = 0\)
Let \(\log_{3} x = a\), then \(\log_{x} 3 = \frac{1}{a}\).
\(a - \frac{3}{a} + 2 = 0\)
\(a^{2} + 2a - 3 = 0\)
\(a^{2} - a + 3a - 3 = 0\)
\(a(a - 1) + 3(a - 1) = 0\)
\((a - 1)(a + 3) = 0\)
\(\text{a = 1 or -3}\)
\(\log_{3} x = 1 \implies x = 3^{1} = 3\)
\(\log_{3} x = -3 \implies x = 3^{-3} = \frac{1}{27}\)