If 27x =93x, find x.

We are given $27^{x}=\frac{9}{3^{x}}$. We have to find the value of $x$

Since $\left(3^{3}\right)^{x}=\frac{3^{2}}{3^{x}}$

By using the law of exponents $\frac{a^{m}}{a^{n}}=a^{m-n}$ we get,

$3^{3 x}=3^{2-x}$

On equating the exponents we get,

$\begin{aligned} 3 x &=2-x \\ 3 x+x &=2 \\ 4 x &=2 \\ x &=\frac{2}{4} \end{aligned}$

$x=\frac{1}{2}$