... sequence that starts at a scale of 2.5% and finishes at 400% after about 5 seconds.
Your images are going to fall apart at 400%. You want to keep your maximum scale close to 100% or, when working with 3D layers keep the distance between the camera and the layer the same or greater than the zoom value of the camera. For either technique the easiest way to adjust the speed of the move is with the graph editor.
It sounds like you would be better off turning your image sequence into a 3D layer and moving in with a camera. The other option would be to use exponential scale from the keyframe assistants. Trying to adjust velocity to get a natural looking move with scale is a bit like herding cats. It's possible, but not very easy. Getting pleasing results with a camera and 3D layer is usually a much better way to go.
The image is a company logo and we are pushing into one of the letters so the screen goes completely white as it is filled, therefore we don't see any quality loss as we don't see the edges past 100% scaling.
You mention using "exponential scale from the keyframe assistants" could you elaborate on what you mean by this. I had already selected exponential scale between the first and last keyframe, however the initial movement is still much too fast.
I was hoping the logo starts scaling up slowly and gets faster as we approach it
I had a read through that section on the Graph editor but I still couldn't work out where I'm going wrong here.
Sorry, I'm sure I'm being completely dense, combined with lack of sleep and some horrific deadlines, leaving me unable to see the wood for the trees.
Should I be able to re-arrange the keyframes to get the desired effect using the graph editor then?
I feel like I should be sat in the corner with the dunce's hat on
If you want to stick with adjusting the scale you can do this two ways. Edit the value graph or edit the speed graph.
The problem is that a camera move, a zoom or a dolly in, is not linear and scale is. This means the graphs are not a good representation of what you'll see visually. This makes them hard to use. At a constant rate for scale the appearance is that the increase in size slows down to closer you get to the final value. In other words, as you scale an object up at a constant rate, the visual appearance is a gradual slowing.
A speed graph edited to look like this gives the appearance of a constant rate scale of the layer when you expect an acceleration at the end. There's just not enough granularity in the graph editor or enough control to predictably achieve the results you want.
You'll have better luck editing the value graph to look something like this:
While this looks extreme, you will get closer to achieving the results you want using the value graph. Once again, the amount of control and the resolution of the graph combined with the visual tomfoolery that scaling an object brings with it makes this a difficult way to achieve predictable results.
You select the graph you'll edit by clicking on the second icon from the left.
As I said before, you'll have much better luck getting the look you want if you make the layer 3D and move a camera toward it.