Given that (log_{3} x – 3log_{x} 3 + 2 = 0), find the values of x.
Explanation
(log_{3} x – 3log_{x} 3 + 2 = 0)
(log_{3} x – 3log_{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})