= Effective cyclic membership in a free product
Suppose $P$ has a soluble <word problem for a group> and $z$ generates an infinite <cyclic group>. In $P*\langle z\rangle$, for $g\ne1$ the words $gz$ and $[g,z]$ are cyclically reduced with two and four syllables respectively. Their nonzero powers have syllable lengths $2|k|$ and $4|k|$. Given an input's reduced <free product> form, its length bounds the possible exponents; compare the finitely many candidates using the base word algorithm. Membership and the exponent in each of these infinite cyclic subgroups are therefore computable, with no algorithm for the order of $g$ required.
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