I just started to play around with the idea of the matrix logarithm. From my research, the simplest way to compute the value of the matrix logarithm is a generalization of the Taylor series of the natural logarithm — you just replace the $1$, with $I$:
$$\ln(A + I) = \sum_{n=1}^{\infty} (-1)^{-n + 1} \frac{A^{n}}{n}$$
Why not define the matrix logarithm, as the generalization of the $\ln(x)$ as the integral of the function $\frac{1}{x}$ instead$?$
$$\ln(x) = \int_{1}^{x} \frac{1}{t} dt = \lim_{n\to\infty} \sum_{i=1}^{n-1}\frac{1}{1 + \frac{(x - 1)i}{n}}\frac{x-1}{n}$$
so why not define the matrix logarithm as such
$$\ln(A) = \int_{I}^{A} \frac{1}{x} dx = \lim_{n\to\infty} \sum_{i=1}^{n-1}\frac{1}{I + \frac{(A - I)i}{n}}\frac{A-I}{n}?$$
I know it's a bit weird that the limits of integration are matrices, but I don't see anything wrong with that, especially because we can just use the Riemann sum directly. Do we avoid using that definition because not every matrix is invertiable$?$ Because the two matrices in this sum don't always commute? or do we do use this definition and my reseasrch was incomplete?