Matching Powers

In 3^x = 81 the unknown sits in the exponent. Start by writing 81 as a power of 3: 81 = 3 × 3 × 3 × 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.