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In the exponential growth formula, $N = N(0)e^{kt}$, I know that $t$ is for time, but what is the unit of measure? Years? Months? Days? Hours?

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Depends on the problem, but generally $t$ is dimensionless in the general formula: $N = N_0e^{kt}$. – William Feb 15 '12 at 20:49
Inside the exponential, that should be dimensionless. So if you want $t$ to be in sec, then $k$ should be in 1/sec. – GEdgar Feb 15 '12 at 21:07

$$\frac{dN}{dt} = kN$$ That is, the rate of the growth of the population is directly proportional to the number at that instant.

The units of k will determine the units for t.

When solved for N, $$\frac{dN}{N} = k.dt$$ $$\int_{N_0}^N\frac{dN}{N} = \int_0^t k.dt$$ $$\ln \frac{N}{N_0} = kt$$ $$N = N_0.e^{kt}$$

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