(This question is originally from Math.SE, where it didn't receive any answers.)

Is there a first-order formula $\phi(x) $ with exactly one free variable $ x $ in the language of ordered fields together with the unary function symbol $ \exp $ such that in the standard interpretation of this language in $\Bbb R $ (where $ \exp $ is interpreted as the exponential function $ x \mapsto e^x $), $\phi (x) $ holds iff $ x=\pi $?

(Since a negative answer to this problem would imply that $e$ and $\pi$ are algebraically independent, I cannot expect anyone to give a complete proof that it isn't possible. However, in the case that one suspects strongly that the answer to this question is negative, I would already be pleased if someone could give intuitive arguments why one shouldn't believe in the definability of $\pi$.

However because there are such intricate connections between exponential and trigonometric functions, I don't think that $\pi$ should be undefinable.)