I know that this matrix $$M=\begin{pmatrix} e^{a_{1,1}} & e^{a_{1,2}} & e^{a_{1,3}}\\ e^{a_{2,1}} & e^{a_{2,2}} & e^{a_{2,3}} \\ e^{a_{3,1}} & e^{a_{3,2}} & e^{a_{3,3}} \end{pmatrix}$$ can't be expressed as $\exp(A)$ where $$A=\begin{pmatrix} a_{1,1} & a_{1,2} & a_{1,3}\\ a_{2,1} & a_{2,2} & a_{2,3} \\ a_{3,1} & a_{3,2} & a_{3,3}\end{pmatrix}$$ In fact, as stated in Wikipedia, the exponential of $X$ is the $n\times n$ matrix given by the power series $$e^{X}=\sum _{k=0}^{\infty }{1 \over k!}X^{k}$$ where $X^{0}$ is defined to be the identity matrix $I$ with the same dimensions as $X$.
Anyway, I would like to know if there is a way to write $M$ as a function of $A$.