A log-Hadamard matrix of a complex Hadamard matrix $H$ is any real matrix $\Phi_H$ satisfying: $$\left[H\right]_{jk}=\exp\left(i\left[\Phi\right]_{jk}\right).$$ The phases $\left[\Phi\right]_{jk}$ may be chosen to belong to $[0, 2\pi)$.
As an example, let us consider the Fourier matrix of size four: $$F_4=\left[\begin{array}{rrrr} 1&1&1&1\\ 1&i&-1&-i\\ 1&-1&1&-1\\ 1&-i&-1&i\end{array}\right] \Longrightarrow \Phi_{F_4}=\pi\frac{2}{4}\left[\begin{array}{cccc} 0&0&0&0\\ 0&1&2&3\\ 0&2&0&2\\ 0&3&2&1 \end{array}\right].$$