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Because a JPEG logo is pixel-based and SVG files are vector-based, you'll need to convert the logo into vectors. You could either do this manually (which is the most work, but would probably yield the best result), or you can use Live Trace to trace it for you. https://helpx.adobe.com/illustrator/using/tracing-artwork-live-trace-or.html
With Live Trace you may not get a very clean result, especially for a logo and if the starter file is low resolution.
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The typeface used in the grey word looks like Optima.
The green thing is just a circle with a gradient (see the documentation about gradients)
The "handwritten" word probably uses also a typeface, but I don't know which one (you could try out www.whatfontis.com). But that one can also be autotraced.
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If you had posted the logo previously I didn't notice the link. After seeing it:
I don't know how this will be used or what control over the logo design you have, but while you're recreating the logo this may be a good time to think about a redesign/update.