Logarithm of transfinite numbers

What value, if any, can be ascribed to the logarithm of aleph_n?

Take log_2(aleph_0) as an example.

log_2(aleph_0) can't be finite, and it can't be bigger than aleph_0, so
it has to equal aleph_0.

But log_2(aleph_0) = aleph_0 implies 2^aleph_0 = aleph_0, whereas in
fact 2^aleph_0 = aleph_1.

Where did it go wrong?

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