Nillie,
Depth of Field is affected by several factors that you can control with the C-7070: aperture size, focal length and distance. A smaller f-stop number or wider aperture gives less DOF. A longer focal length or zoom telephoto gives less DOF. A closer distance shortens DOF too.
To get a short depth of field try these techniques. Set the mode to Aperture priority, A mode. Lower the f-stop number to the smallest number (bigest aperture opening) for the zoom setting. Check the shutter speed and shoot at your smallest f-stop.
At 27 mm wde angle that's f/2.8. At 110 mm telephoto the smallest aperture is f/4.8. Either zoom will yield similar results because as you zoom in (longer focal length) DOF decreases. As you zoom in the aperture may not change (think of the iris opening size) but the f-stop number is getting larger - f-stop number is just the ratio of aperture to focal length.
Also move as close to your subject as you can to get the composition you're looking for. Set the zoom to fit the frame you want from your closest point. At close distances DOF decreases, giving you the blurred background you're after.
For a human face you want to zoom in to the 50 mm range or higher to avoid distortion. With the C-7070 that's just about 2x zoom or near 1/4 of the zoom scale or higher.
So set the aperture to the smallest setting, move in close and zoom in as far as you can. These 3 factors will reduce DOF.
The C-5050 achieves shorter DOF with a wider f-stop, f/1.8. The C-8080 has a larger sensor but achieves short DOF with a smaller f-stop, f/2.4 or a longer zoom, up to 140 mm.
The DOF calculator linked in one of the posts above
http://www.dofmaster.com
will give you a DOF calculator you can use to create a chart of DOF. You can create the chart to show DOF for any given f-stop, distance and focal length. You can print it out and use the chart to predict where you'll get short DOF, or conversely, how you can capture the entire scenic landscape with long DOF, which is what the C-7070 is so good at.
Ed posted a DOF comparison above. That suggests the only way you'll get shorter DOF with a DSLR is with a big aperture lens or long zoom lens. That is, a lens with an f/1.4, f/2 or f/2.8 or a telephoto over 150 or 200 mm. An f/2.8 is back near the C-7070 again. You can add the teleconverter lenses to the C-7070 and get a longer zoom; theTCON 17C or the TCON 30C I think they're called, to get either 187 mm or 330 mm.
Dave
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UW Photographer w/ C-8080