First slide

Types of real valued functions XI

Question

For $a > 0, \neq 1$ the roots of the equation $\log_{ax} a + \log_{x} a^{2} + \log_{a^{2} x} a^{3} = 0$ are given by

Moderate

A

$a^{- 4 / 3}$
B

$a^{- 3 / 4}$
C

a
D

$a^{- 1 / 2}$

Solution

$\log_{ax} a + \log_{x} a^{2} + \log_{a^{2} x} a^{3} = 0$
$\begin{array}{l} or \frac{1}{\log_{a} ax} + \frac{2}{\log_{a} x} + \frac{3}{(\log_{a} a^{2} x)} = 0 \\ or \frac{1}{1 + \log_{a} x} + \frac{2}{\log_{a} x} + \frac{3}{(2 + \log_{a} x)} = 0 \end{array}$
$Let \log_{a} x = y, we have \frac{1}{y + 1} + \frac{2}{y} + \frac{3}{2 + y} = 0$
$\begin{array}{ll} or 6 y^{2} + 11 y + 4 = 0 \\ or y = \log_{a} x = - \frac{1}{2}, - \frac{4}{3} \\ \Rightarrow x = a^{- 4 / 3}, a^{- 1 / 2} \end{array}$

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