Appropriate link-function in a Poisson Model

does somebody know why the log-link function is always assumed to be the "standard" link function in Poisson models? E.g. regarding count data (such as numbers of individuals), under which circumstances does it make sense to assume that the response variable increases exponentially with a covariate? I can imagine situations where a linear, a sigmoidal or a gaussian-like dependence of the outcome on the covariate makes sense, e.g. if we consider number of individuals depending on an environmental variable, but are there many real-life examples for an exponential depedency? Thanks.