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## Main Question or Discussion Point

I'm failing to understanding why e shows up in continuous growth and decay. I've read the BetterExplained article, watched the KhanAcademy videos and read about it in Morris Kline's calculus book (which gives a somewhat better explanation than most for intuition).

I think I get it in terms of compounding interest - say if you have $1 which gains 100% interest over some period t and it's compounded at every instant (so 100/n is the % gain at each instant), then the amount you'll have at the end of the period is e.

But I can't visualize this for other things, for example, this question:

The intensity L(x) of light x feet beneath the surface of the ocean satisfies the differential equation dL/dx=kL. The intensity of light cuts to half at 18ft. You cannot work without without artificial light if the intensity falls below 1/10th of the surface value. About how deep can you expect to work without artificial light?

The computation was easy enough, but I don't understand the meaning of the expression I got L=exp(1/18ln(0.5)x)

I think I get it in terms of compounding interest - say if you have $1 which gains 100% interest over some period t and it's compounded at every instant (so 100/n is the % gain at each instant), then the amount you'll have at the end of the period is e.

But I can't visualize this for other things, for example, this question:

The intensity L(x) of light x feet beneath the surface of the ocean satisfies the differential equation dL/dx=kL. The intensity of light cuts to half at 18ft. You cannot work without without artificial light if the intensity falls below 1/10th of the surface value. About how deep can you expect to work without artificial light?

The computation was easy enough, but I don't understand the meaning of the expression I got L=exp(1/18ln(0.5)x)