  Let $G$ be an acylindrically hyperbolic group and $E$ an exponential equation
over $G$. We show that if $E$ is solvable in $G$, then there exists a solution
whose components, corresponding to loxodromic elements, can be linearly
estimated in terms of lengths of the coefficients of $E$. We give a more
precise answer in the case where $G$ is a relatively hyperbolic group. Under
some assumption of general character, the solvability and the search problems
for exponential equations over $G$ can be reduced to the peripheral subgroups
of $G$.
