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So, sometimes Mathematica cannot evaluate the base-0 logarithm of certain matrices.

But we know that the expressions $0^x$ and $\log_0 x$ are always equivalent. What can I do so that Mathematica always would interpret Log[0, x] as 0^x? For instance, Log[0, 0]=1, but not only that.

Another example,

MatrixFunction[Log[0, #]&, {{1/2,1/2},{1/2,1/2}}]

should give result {{1/2,-1/2},{-1/2,1/2}}.

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The following code seems to work:

Unprotect[Power]; Power[0, 0] = 1; Protect[Power];
Unprotect[Log]; Log[0, x_] = Power[0, x]; Protect[Log];
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