Matching Powers

In ${3}^{x} = 81$ the unknown sits in the exponent. Start by writing 81 as a power of 3: $81 = 3\cdot 3\cdot 3\cdot 3$, which is ${3}^{4}$. Now the equation reads ${3}^{x} = {3}^{4}$, the same base on both sides.

Equal powers of the same base have equal exponents, so x = 4. The reason is the shape of the graph: an exponential only ever rises (or only ever falls), so it never reaches the same height twice. If two powers of 3 are equal, their exponents must be equal too.

The bases must match first: in ${3}^{x} = {2}^{4}$ you cannot set x equal to 4, because 3 to the 4 is 81, not 16.