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1/2 = 0.5

Work Shown:

3/4 = 0.75

Gaining 3/4 of its size means that it is 1.75 times larger. For example, if the population starts at 100 members, then gaining 3/4 = 75% of the population means it bumps up to 175.

If we start at 300 members, then it jumps to 300*1.75. I won't multiply that out as you'll see why later below.

N(t) = 300*(49/16)^t

300*1.75 = 300*(49/16)^t ... replace N(t) with 300*1.75

1.75 = (49/16)^t the 300s cancel after dividing both sides by 300

(49/16)^t = 1.75

Log[ (49/16)^t ] = Log[ 1.75 ]

t*Log(49/16) = Log(1.75)

t = Log(1.75)/Log(49/16)

t = 0.5

It takes half a week for the population to gain 3/4 of its size. This is true no matter what the starting point is.