The time it takes a colony that has been growing exponentially to double its numbers (doubling time) is longer than the time it takes a single cell to go from division to division (generation time)
they are different. The time it takes a colony that has been growing exponentially to double its numbers (doubling time) is longer than the time it takes a single cell to go from division to division (generation time) because more of the cells are the ones that have just divided than the ones that are about to divide. Mathematically, N=N0*2^(t/td) = N0*e^(t/tg) --> td=log(2)*tg
Both are same in a very special case where we consider
1) a system where cells grow as 1->2->4->8->16->32...and so on
2) where we ignore the cell death
However, In other cases, both will be different as doubling time only considered as the time taken by cells to become double in number ignoring its growing fashion (e.g. 1->3->9->27->81->243...and so on). Also, if there is significant cell death, generation time will not be affected by cell death but doubling time will definitely be affected by it.