resizing from big format to a small logo will not make it losing sharpness...what u mean is resizing from small to big.
And what u r saying about the dpi is not correct. Dpi is a value for printing and has nothing to do with the digital picture ...what u mean is ppi pixels per inch...and the ppi value alone would not change the size of a picture...only the the dimensions (length x height) or the resolution (total pixels in lenght x height) plus the ppi does change the size of a picture...so u always need 3 of those values to change the actuall size of a digital picture.
e.g. u can have a 1inch(or cm) x 1inch(or cm) picture with either 100 or 1000ppi and u can also have a 1000inch x 1000inch picture with 100 or 1000ppi or even a 1000inch x 1000inch with only 1ppi.
what makes the picture sharp is the right compination out of those 3 values, therefore what makes a digital picture smaller or bigger is only the resolution or the dimensions itself. the ppi vallue can or most of the time will stay the same...even if u would not need 300 ppi in a 1inch x 1inch picture to make it look sharp because thats pretty much the size of a desktop icon ^^
another quick example is your monitor lets say it has ~24 inch(60 cm) in length with total pixel of 1920 at that side...that makes it 80 pixels per inch (80 ppi) and i would say that monitor is really sharp. so if you do a logo the size of this monitor and with a ppi value of 300 it should be more than enough for all the porposes we need it for...as you would never need a larger logo as the monitor is and even then u would only need a 80 ppi logo to let it appear sharp so its nearly 4 times as much as u need minimum.
Maybe that helps some of the other guys too