What does the halting problem mean for a 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.