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  4. The equation log3(3x-8)=2-x has the solution

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Q. The equation $log_{3}\left(3^{x}-8\right)=2-x$ has the solution

Linear Inequalities

Solution:

$log_{3}\left(3^{x}-8\right)=2-x$
$ \Rightarrow 3^{x}-8=3^{2-x},$ where $3^{x}-8 > 0$
$\Rightarrow 3^{x}-8=\frac{9}{3^{x}}$
$ \Rightarrow \left(3^{x}\right)^{2}-8(3^x)-9=0$
$\Rightarrow \left(3^{x}+1\right)\left(3^{x}-9\right)=0$
$ \Rightarrow 3^{x}=9 \left[\because 3^{x}\ne-1\right]$
$\therefore x=2 $