Scaling a glyph without changing the stoke weight... nope. But there are some workarounds. Reading between the lines of your question, it sounds like you've identified the typeface and have it. Unfortunately, you haven't said what it is, but I'm going to assume it's a sans if Case 1 doesn't solve your problem.
Case 1: The logotype was created with simple Cap and Small Caps.
Case 2: The font doesn't have small caps.
Create your logotype as above in Illustrator. Scale the "S" as needed.
Convert to outlines (Ctl-Shift-O/Cmd-Shift-O).
Deselect (Cmd/Ctl-Shift-A or simply click on a blank area of the canvas)
This example uses Myriad Pro, which lacks small caps (most sans serifs do):
Select the S (black arrow tool).
Choose Object > Path > Offset Path. Turn on Preview so you can see what's going on.
Here you'll have to experiment a bit. In this example, I found a 2 pixel (point) offset worked.
- In the Layers panel, twirl open your Layer 1 and click the eyeball by the lower of the two S shapes to turn it off.
- Adjust the spacing (kerning) as needed. Each of your characters is now a vector outline, so you can select them individually and tweak to taste.
Case 3: Nope. It's a serif, and there are no small caps.
- Follow the steps above. You will have to adjust the serif paths by hand, selecting individual points with the white arrow tool.