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  4. The product of all the real solution(s) of the equation 2 log 9(x-1)=2+ log (x-1)2 3- log √3(x-1) is

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Q. The product of all the real solution(s) of the equation $2 \log _9(x-1)=2+\log _{(x-1)}^2 3-\log _{\sqrt{3}}(x-1)$ is

Complex Numbers and Quadratic Equations

Solution:

Let $t =\log _3( x -1)$
Hence, $t=2+\frac{1}{t^2}-2 t \Rightarrow 3 t=\frac{2 t^2+1}{t^2}$
$ \Rightarrow 3 t^3-2 t^2-1=0 \Rightarrow(t-1)\left(3 t^2+t+1\right)=0$
Hence, $t=1 \Rightarrow \log _3(x-1)=1 \Rightarrow x=4$