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  4. The equation log 3(3-x)- log 3(3+x)= log 3(1-x)- log 3(2 x+1) has

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Q. The equation $\log _3(3-x)-\log _3(3+x)=\log _3(1-x)-\log _3(2 x+1)$ has

Continuity and Differentiability

Solution:

We have $\log _3(3-x)-\log _3(3+x)=\log _3(1-x)-\log _3(2 x+1)$
$\Rightarrow \log _3((3-x)(2 x+1))=\log _3((3+x)(1-x)) \Rightarrow(3-x)(2 x+1)=(1-x)(3+x)$
$\Rightarrow 6 x-2 x^2+3-x=3+x-3 x-x^2 \Rightarrow -2 x^2+5 x+3=-x^2-2 x+3 \Rightarrow x^2-7 x=0$
$\Rightarrow x=0,7$
But, $x=7$ (rejected)
So, $x =0$ is the only solution.