Existence of logarithm of non-singular matrix From matrix function point view, we can define logarithm of a non-singular matrix as some series such like Taylor series:
$$\log(I+N)=\sum_{i=1}^\infty{(-1)^{i+1}N^i \over i}$$ where $\|N\|<1$.
Here I want to construct a ODE which connected the log and equation.
For example, for exponential matrix, we study following equation:
$$X'(t)=AX(t).$$
which can be used to study its solution's existence and uniqueness.
For log, do we have such kind functions？
 A: @ Benjamin , your posts are unclear. 1. There are $2$ mistakes in your definition of $\log(I+N)$. 2. You speak about " ODE which connected the log and equation"; in fact you do not really want to derive the $\log$ or $\exp$ function; you want to derive the functions $t\rightarrow \log(tA)$ or $t\rightarrow \exp(tA)$, that is much easier. 3. To hope the uniqueness of the $\log$ is vain.
You can define the principal logarithm of any complex matrix $A$ that has no eigenvalues in $(-\infty,0]$ cf. my post in Roots and Logarithms of Matrices.
For this definition, if $A$ is real, then $\log(A)$ is real and moreover, for every such complex matrices,  $\log(A^{1/2})=1/2\log(A)$ and $\log(A^{-1})=-\log(A)$.
As Yves wrote, one has:
Proposition. Let $A\in M_n(\mathbb{C})$. Then, for convenient values of $t$,  $\log(I+tA)'=A(I+tA)^{-1}$.
Proof. 1. Assume that $||tA||<1$. The result is clear, deriving the series above. 2. The RHS and LHS are holomorphic functions. Using the theorem of "extension of equalities", we obtain the required result for all convenient complex $t$.
Finally we obtain the ODE $\log(X)'=AX^{-1}$ or $Y'=Ae^{-Y}$.
EDIT. Answer to Benjamin. When you consider the principal $\log$, one has $\exp(\log(X))=X$; let $A\in M_n(\mathbb{R})$ be s.t. its eigenvalues are not in $(-\infty,0]$ and let $m$ be the max of positive eigenvalues (if there exist, otherwise $m=0+$); then the function $Y:t\in (-1/m,+\infty)\rightarrow \log(I+tA)$ satisfies (1) $Y'=Ae^{-Y},Y(0)=0_n$. Conversely, (1) admits locally a sole solution, the one above.  
A: I founded some useful proof of the existence and uniqueness of logarithm of matrix here.
Reference the conclusion:
Theorem 3.4. Let $A$ be a real $n\times n$ matrix and let $(X-\alpha_1)^{r_1},\ldots,(X-\alpha_m)^{m_1}$ be its list of elementary divisors or, equivalently, let $J_{r_1}(\alpha_1),\ldots,J_{r_m}(\alpha_m)$ be list of Jordan blocks. Then, $A$ has a real logarithm iff $A$ is invertible and if, for every $r_i$ and every real eigenvalue $\alpha_i<0$, the number, $m_i$, of Jordan blocks identical to $J_{r_i}(\alpha_i)$,is even
