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De Morgan's Law

Solve the fol...

Question

Solve the following equation:
${log}_{3} (3^{x} - 8) = 2 - x$

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Solution

${l o g}_{3} (3^{x} - 8) = 2 - x \Rightarrow 3^{x} - 8 = 3^{2 - x} \Rightarrow 3^{x} - 8 = \frac{9}{3^{x}} \Rightarrow {(3}^{x})^{2} - {8.3}^{x} - 9 = 0 \Rightarrow (3^{x} - 9) (3^{x} + 1) = 0$
$\Rightarrow 3^{x} \neq - 1$ where $3^{x}$ is always positive.
$\Rightarrow 3^{x} = 9 \Rightarrow x = 2.$

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