My students have never formally worked with geometric or arithmetic sequences before and I thought that I might use this problem to introduce them to those ideas. But as my ideas for the lesson took form, I decided that I wanted them to play with the mathematics and that too much formality might squash that. Here are the 7 steps that were part of that discovery.
Here are two main items that my students noticed. For some reason I was so focused on the relationship between the log of a geometric sequence and the resulting arithmetic sequence that I didn't notice the very first thing that my students noticed! (Note: The bases of 2 and 4 were chosen to make it easier for students to make observations.)
Here is what my students wondered:
1) Is the ratio of f(x):g(x) always 2?
2) What if a and r weren't even, positive integers?
3) What happens if 0 < r < 1?
Here are screenshots from Desmos to show some of the student's investigations.