What logging does is create an economic incentive to plant the trees and let them grow. The problem with algae is that it doesn't have economic value currently. People dream about turning it into food or biofuels but that's not currently viable.

Although if you were only interested in sequestering carbon (which currently has close to 0 economic incentive) you could grow algae, filter them out of the water, and then pump them deep underground into old gas/oil wells. It's still a lot of energy but possibly more viable than most carbon capture proposals. Over millions of years that algae will probably turn back into coal/oil.

Also, farming algae in natural waterways tends to have its own environmental impacts.