Compute the logarithm of a matrix

There have already been challenges to compute the exponential as well as sine& cosine of a matrix. This challenge is about finding the (natural) logarithm of matrix.

You task is to write a program of function that takes an invertible \$n \times n\$ matrix \$A\$ as input and returns the matrix logarithm of that matrix. Like the real logarithm the matrix logarithm of \$ A\$ is defined to be a matrix \$L\$ with \$ exp(L) = A\$.

Like the complex logarithm the matrix logarithm is not unique, you can choose to return any of the possible results for a given matrix.

Examples (rounded to five significant digits):

log( [[ 1,0],[0, 1]] ) = [[0,0], [0,0]]
log( [[ 1,2],[3, 4]] ) = [[-0.3504 + 2.3911i, 0.9294 - 1.0938i], [1.3940 - 1.6406i, 1.04359 + 0.75047i]]
log( [[-1,0],[0,-1]] ) = [[0,pi],[-pi,0]] // exact
log( [[-1,0],[0,-1]] ) = [[0,-pi],[pi,0]] // also exact
log( [[-1,0],[0,-1]] ) = [[pi*i,0],[0,pi*i]] // also exact

log( [[-1,0,0],[0,1,0],[0,0,2]] ) = [[3.1416i, 0, 0], [0, 0, 0], [0, 0, 0.69315]]
log( [[1,2,3],[4,5,4],[3,2,1]] ) = [[0.6032 + 1.5708i, 0.71969, -0.0900 - 1.5708i],[1.4394, 0.87307, 1.4394],[-0.0900 - 1.5708i, 0.71969, 0.6032 + 1.5708i]]

If you want to try out more examples use the function digits 5 matrix logarithm followed by a matrix in Wolfram Alpha

Rules:

• You can Input/Output matrices as nested lists
• You can Input/Output complex numbers as pairs of real numbers
• You can assume the logarithm of the input number exists
• Your result should be accurate up to at least 5 significant digits
• You only have to handle matrices of sizes \$2\$ and \$3\$
• You program may return different results when called multiple times on the same input as long as all of them are correct
• This is code-golf the shortest solution (per language) wins

code-golf math matrix

Meta:Compute the logarithm of a matrix

• Is this a duplicate (I could only find the two liked challenges) ?
• Is my explanation clear ?
• Is the task to complicated for code-golf (the standard approach requires calculating the Jordan decomposition of a matrix) ?
• Should I restrict the challenge to diagonalizable matrices?