I'd like to make this logo small enough to use on a Facebook profile photo which is 160 × 160 px. Will I have to redesign it to 160 × 160px?
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While you're in the adobe illustrator file, select all the elements of the logo and up on the toolbar go
Object > Expand or
Object > Expand Appearance and have the options
Stroke checked if available.
Once that's done you can resize it to what ever size you want without losing any quality. The stroke and fill ratios will stay the same. This technique is much faster than any other.