Assume the world consists of two regions, the North and South with the following production functions and technology lags in the south by $\tau$ years:
$$ Y_N = A_N (t) (1-a_L) L_N; \; \; \dot{A}_N = a_L L_N A_N (t)$$
$$ Y_S = A_S (t) L_S ; \; \; A_S = A_N (t - \tau) $$
The growth rate of output per worker in the North is $3 \% \; year^{-1}$ and if $a_L \simeq 0$, what $\tau$ must be for output per worker in the North to exceed that in the South by a factor of 10?
For output per worker in the north to exceed that in the south by a factor of ten:
$$ 10 = \frac{Y_N/L_N}{Y_S/L_S} = \frac{A_N(t)(1-a_L)}{A_S(t)} $$
with $a_L \simeq 0$ this goes to
$$ \to \frac{A_N (t)}{A_N (t-\tau)}$$
My book says this is equal to $e^{0.03 \tau}$
Could someone explain where this last result comes from please?