Answer
$x=-4$
Work Step by Step
The given equation is
$\Rightarrow 27^{x}=9^{x-2}$
Rewrite $27$ as $3^3$ and $9$ as $3^2$.
$\Rightarrow (3^3)^{x}=(3^2)^{x-2}$
Use $(a^n)^m=a^{n\cdot m}$
$\Rightarrow 3^{3(x)}=3^{2(x-2)}$
Simplify.
$\Rightarrow 3^{3x}=3^{2x-4}$
Equate the exponents.
$\Rightarrow 3x=2x-4$
Subtract $2x$ from each side.
$\Rightarrow 3x-2x=2x-4-2x$
Simplify.
$\Rightarrow x=-4$
Check: $(x=-4)$
$\Rightarrow 27^{x}=9^{x-2}$
$\Rightarrow 27^{-4}=9^{-4-2}$
$\Rightarrow 27^{-4}=9^{-6}$
$\Rightarrow \frac{1}{27^{4}}=\frac{1}{9^{6}}$
$\Rightarrow \frac{1}{531441}=\frac{1}{531441}$
True.
Hence, the solution is $x=-4$.
