What does the halting problem mean for a Babbage machine? I read that the Babbage machine is Turing complete. Which means that no decision Turing machine will halt on the question "does the Babbage machine computes the logarithms of its input?" (for example).

The way I imagine that is there is no finite method (i.e. algorithm) which would examine the Babbage machine, possibly by testing it, but most probably by dismantling it like a watch and concludes that it indeed computes logarithms.

Would that be a correct way of thinking the halting problem on the Babbage machine?
No. The undecidability of the halting problem for a particular model of computation means that there's no algorithm that takes as input a description of an arbitrary machine&nbsp;$M$ in that model and an input&nbsp;$x$ and determines whether $M$&nbsp;halts when run with input&nbsp;$x$.

As a consequence, there's no algorithm that takes as input a description of a machine and determines whether that machine computes logarithms. However, that doesn't mean that you can't prove that a specific machine computes logarithms. Indeed, people do this all the time, as exemplified by Knuth's famous quote, &ldquo;Beware of bugs in the above code; I have only proved it correct, not tried it&rdquo; (emphasis mine). Proving the code correct requires proving that it terminates and computes the thing it's supposed to compute.