Type of Distribution Suppose a data set looks like exponential growth. See this. Would you use an exponential distribution to model this?
 A: Exponential growth and the exponential distribution are different things. 
If I thought the data was generated by a process that undergoes exponential growth I would think of a model like:
$$
population = ae^{ct}
$$
Where t is time, a is the population at time 0, c is the exponential growth rate and e is the natural log's base.
In order to shoehorn this into a linear regression setup I would take the log of both sides. and see that:
$$
log(population) = log(a)+ ct
$$
So then I would regress log(population) on t. Then exponentiate the intercept and call that the population at time 0. and the coefficient on t as the exponential rate. There are problems with this approach.
A: No, the exponential distribution has lots of smaller values and fewer larger values.  It is typically for things like the waiting times between random events that are equally likely to happen any time.
What you are talking about is a particular type of non stationarity in your time series.  Typically you would seek to remove thus exponential growth before doing your analysis.  Taking logs could be one way of bringing the data back to linear growth that is easier to work with.
