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the basis of exponential growth is that the reproduced stuff embeds the FULL growth capacity of that which is being reproduced. there's a no-loss fractal embedding. it's creating a "perfect" copy that is capable of making further perfect copies

every other form of exponential growth is just a multiplier on the base 2 example above e.g. if instead of making one copy of urself u can make 2 copies then you start with 1, then 3, then 9 and so on. 3^x

where the magical 2.718 number comes in is when you leave the discrete world when you make a copy you are creating an instant copy of yourself. u go from from 1 -> 2

however if you have 1000 dollars in a bank account with 100% annual interest given monthly, it doesn't go from 1000 -> 2000 instantly. as soon as u get each month's payment, it can immediately start to grow as well. e.g. (100/12 = 8.3% monthly)

intuitively you will sense that with this compounded growth, you will get MORE than x2 (obviously) but probably less than x3 and that's correct. the more you break down the payments to allow compounding to happen ASAP the more u approach the limit of 2.718...

the above example is 100 * 1.083 ^ 12 which is 260 instead of 100 -> 200 you get 260 with daily interest the daily rate is 100/365 = 0.2739 so: 100 * 1.002739 ^ 365 = 271.38... which is pretty close to the 2.718.... multiplier

u can calculate to infinite precision because technically your dollar is made up of pennies and those pennies can be subdivided into milli-pennies that can start compounding themselves and those into micro pennies etc... but in practice no one does this

natural things however DO do this a plant w/ 1 square centimeter of surface area "1 dollar" actually contains millions of cells. it's like 1 dollar divided into millions of pieces, each reproducing itself this is why e growth is "natural"

the "natural" inclination is self replication and that replicant contains the ability to self replicate as well. so natural growth is e^x at a systems level even though at the micro level it is 2^x

the above is just speculation though going back to the 2.718 as *THE* natural growth rate, it also explains why the derivative of e^x is also e^x a derivative basically says "what is the rate of change along this function"

when you ask the natural growth function "at what rate are you growing at?" it'll reply "i'm just being natural ME!" that is... at any given snapshot of a natural growth situation that snapshot itself wholly contains enough information to calculate the growth rate