Discrete logarithm problem

Definition
This problem usually makes sense in the context of an encoding of a group or a multi-encoding of a group. The problem asks for the following: given the code-words for elements $$g,h$$ of the group such that $$h$$ is a power of $$g$$, find any integer $$x$$ such that $$g^x = h$$.

In the context of a black-box cyclic group, this problem is typically asked with $$g$$ chosen to be a generator of the group.

Caveat
Note that $$x$$ is unique modulo the order of $$g$$. Usually, the problem is discussed in a context where the order of $$g$$ is known, i.e., we already know the answer to the element order-finding problem for $$g$$. Given this information, finding one value of $$x$$ is equivalent to finding all values of $$x$$.

However, the problem makes sense even when the order of $$g$$ is not known. In this case, an algorithm that gives any $$x$$ (not necessarily the smallest positive value) suffices.