If x only appears through \log(x), e.g., there are no other constraints of the form x + y \leq 1, then you can write z = \log(x) \geq 0 and replace all occurrences of \log(x) by z \geq 0.
Here, t \geq (\log x)^2 becomes t \geq z^{2}, which is convex in the t, z space.
Solve the resulting problem w.r.t z, and recover x = \exp(z) \geq 1.