prove it if you can

This is an interesting question but not that easy to be proved. The statement is that an irrational number again with an irrational power can be rational. It is very easy to point out at the numbers in natural logarithm table developed to the base 'e' which is irrational and all the values except log1 are irrational. But, can you prove this using simple algebra rather than an example. What I need is a concrete proof, which one can write even without the logarithm table. The proof should be also convincing. Statements such as 'the summation of this series is irrational since each term adds a new digit to the actual value' will not at all be considered since they have no mathematical certainty.
.