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