Rewriting the Base

In ${27}^{x} = 81$, 81 is not a power of 27. But 27 and 81 are both powers of 3: $27 = {3}^{3}$ and $81 = {3}^{4}$. Rewrite both sides with base 3 and the equation becomes one you can match.

Writing 27 as ${3}^{3}$ turns ${27}^{x}$ into ${\left({3}^{3}\right)}^{x}$, and a power of a power multiplies the exponents: that is ${3}^{3x}$. So the equation reads ${3}^{3x} = {3}^{4}$. Equating the exponents gives $3x = 4$, and $x = \frac{4}{3}$. The answer does not have to be a whole number: a fraction is fine.