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How would I go about changing the first object shape in to the second? At the moment I just seem to be going around in circles with subdivide and smooth but not actually achieving the desired result.

This is Blender 2.76 on Windows

reference image

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From an 'L' shape, I would remove the top and bottom faces, use bevel ⎈ CtrlB to get the two round corners, use the MW Wheel to add some segments, you only want a couple of segments on the curves or you'll have too much work filling in the top.

add rounded edges

Then fill the top with an n-gon face F and use inset I to get a clean edge loop around the top.

top with an inset

Then extend the edges from the curves to the outside to get an all quad mesh. Some more loop cuts can be made through the larger sections to get a more consistent face sizing over the mesh.

extend curve edges

Now a subsurf modifier can be added to get a more rounded finish. If you want the outer corners to stay sharp you can add a crease to them ⇧ ShiftE and add an edge split modifier.

model with modifiers

Here is an example to look at

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