  According to [5] we define the $*$-exponential of a slice-regular function,
which can be seen as a generalization of the complex exponential to
quaternions. Explicit formulas for $\exp_*(f)$ are provided, also in terms of
suitable sine and cosine functions. We completely classify under which
conditions the $*$-exponential of a function is either slice-preserving or
$\mathbb{C}_J$-preserving for some $J\in\mathbb{S}$ and show that $\exp_*(f)$
is never-vanishing. Sharp necessary and sufficient conditions are given in
order that $\exp_*(f+g)=\exp_*(f)*\exp_*(g)$, finding an exceptional and
unexpected case in which equality holds even if $f$ and $g$ do not commute. We
also discuss the existence of a square root of a slice-preserving regular
function, characterizing slice-preserving functions (defined on the
circularization of simply connected domains) which admit square roots. Square
roots of this kind of functions are used to provide a further formula for
$\exp_{*}(f)$. A number of examples is given throughout the paper.
