Answer
$\left\{\dfrac{4}{3}\right\}$
Work Step by Step
Expressing both sides of the given equation, $
27^x=81
$, in the same base results to
\begin{align*}\require{cancel}
\left(3^3\right)^x&=3^4
\\
3^{3x}&=3^4
&(\text{use }\left(x^m\right)^n=x^{mn})
.\end{align*}
Since $x^m=x^n$ implies $m=n$, the equation above implies
\begin{align*}\require{cancel}
3x&=4
.\end{align*}
Using the properties of equality, the equation above is equivalent to
\begin{align*}\require{cancel}
\dfrac{\cancel3x}{\cancel3}&=\dfrac{4}{3}
\\\\
x&=\dfrac{4}{3}
.\end{align*}
Hence, the solution set of the equation $
27^x=81
$ is $
\left\{\dfrac{4}{3}\right\}
$.
