First slide

Theory of equations

Question

For a > 0, 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

$a^{1 / 2}$
D

$a^{- 1}$

Solution

We have,
$\frac{\log_{a} a}{\log_{a} a + \log_{a} x} + \frac{2 \log_{a} a}{\log_{a} x} + \frac{3 \log_{a} a}{2 \log_{a} a + \log_{a} x} = 0$
$\begin{aligned} \Rightarrow \frac{1}{1 + t} + \frac{2}{t} + \frac{3}{2 + t} = 0 (let \log_{a} x = t) \\ \Rightarrow \frac{2 t + t^{2} + 2 t^{2} + 6 t + 4 + 3 t^{2} + 3 t}{t (1 + t) (2 + t)} = 0 \\ \Rightarrow 6 t^{2} + 11 t + 4 = 0 \\ \Rightarrow 6 t^{2} + 8 t + 3 t + 4 = 0 \\ \Rightarrow \log_{a} x = - \frac{1}{2}, - \frac{4}{3} \\ \Rightarrow x = a^{- 1 / 2}, a^{- 4 / 3} \end{aligned}$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App