Theory of Equations 1 Question 54
55. Solve $2 \log _x a+\log _{a x} a+3 \log _b a=0$, where $a>0, b=a^{2} x$
(1978, 3M )
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Solution:
- The given equation can be rewritten as
$$ \begin{array}{cc} & \frac{2}{\log _a x}+\frac{1}{\log _a a x}+\frac{3}{\log _a a^{2} x}=0\left[\because b=a^{2} x \text {, given }\right] \\ \Rightarrow & \frac{2}{\log _a x}+\frac{1}{1+\log _a x}+\frac{3}{2+\log _a x}=0 \\ \Rightarrow & \frac{2}{t}+\frac{1}{1+t}+\frac{3}{2+t}=0 \text {, where } t=\log _a x \\ \Rightarrow & 2(1+t)(2+t)+3 t(1+t)+t(2+t)=0 \\ \Rightarrow & 6 t^{2}+11 t+4=0 \\ \Rightarrow & (2 t+1)(3 t+4)=0 \\ \Rightarrow & t=-\frac{1}{2} \text { or } \quad-\frac{4}{3} \\ \therefore & \log _a x=-\frac{1}{2} \quad \text { or } \quad \log _a x=-\frac{4}{3} \\ \Rightarrow & x=a^{-1 / 2} \\ \text { or } & x=a \end{array} $$
