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let, A^i = e i = 1/lnA
it will be the same nevertheless, however I am not sure about K and C
I´m just wondering if e is such a magical number to appear here and there or if its artificial and we could use a different set of numbers to express the same thing/physical law.
i think e is quite an special number <-- especially when rate of change depends on initial value. also it has some special property (base of natural log), i think it is best to leave things with e's
also stretching or compressing exponential function gives e at some point
Hmmm what is so special about e? I only know that the derivative of e^x is again e^x which is quite nice and that I can rewrite e^(ix) in terms of sine and cosine but is there even more to e?
LOL ... not really sure!!
xD I thought there might be some additional stuff I might not know about.^^
Medals 0 You can do it by writing e as a^(1/ln(a)) and apply exponential identities but why would you want to? e is such a nice number :)
e's the best ... LOL
From a practical point of view, if you need to differentiate or integrate your function down the road, would you rather deal with e^x or A^x?
@beginnersmind I was wondering if the appearance of e in some laws of physics was because somewhat had the hearts for e or because there is no other way to state that law. Might have been just some physicist who loved e and we could rewrite some laws. Ok than I guess e really is that great. :)
comes naturally from \( \int 1/x dx \)
f I integrate or differentiate I am always happy to encounter e ;) But if I just add and multiply I could live without e ;) So it depends on the field.
well, a lot of laws are actually of the form e^Cx, so I'm not sure e is especially preferred by nature in those cases. It seems to be notational .
So I could easily rewrite y=K*e^(Cx) with some other base?
i've usually encountered in decay equation and distribution function. decay equation <--- dN/dt = N <-- depends on initial value distribution function --> didn't understand
I started wondering while looking at the decay equation like 10 Minutes ago.^^
@experimentX in the decay function the choice of the base is somewhat arbitrary. Say you have f(t)=e^(-Ct). You could just as easily write f(t)-2^(-Kt), with a different constant.
why ... we choose it as natural base for log while integrating <--- must be some reason.
I'm saying e^(-Ct) and 2^(Ct/ln2) are the same function. They take the same values.
2^(-Ct/ln2) that is
But If you rewrite it using the ln than e is still in there (in that function). So you can rewrite in a way that you can not see e but it is hidden in the ln.
we could use the same logic everywhere, the point is why e so that there is no logs on the power?? 1/ ln 2 ??
Well, there's a constant, which is an experimentally determined number. When you use e as the base the constant is C. When you use 2 it's K=C/ln2. If you actually measure half-life you're measuring K, so ln2 isn't really "hidden" there.
Ops right. ln2 is just some number - there is not an e hidden. So we could really rewrite equations that fit that pattern.
not really sure if i am understanding ..:(
@experimentX Which part? You wrote the same thing as beginnersmind with the rewriting 1/lnA or now 1/ln2.
e^x/nothing <--- why nothing in this case?? must be some special property of e A^x/lnA
Why e^x/nothing? Who wrote that where?
usually equation comes that way when we integrate 1/x dx Oo, i think i need to review decay equation, since i ignored \( \lamda \) factor completely.
LOL ... seems like only e is nice to deal with,
:) I guess than everything is alright. What a nice and interesting discussion. If you follow mathematics and physics in class it seems that e is mysteriously everywhere but apparently it is that way because some people are secretly working for e and we could write it in a different way. :) Nevertheless e is a great number for calculus.
yeah ... that i agree!! makes nice, easy and clean.
To be fair, there are situations where e appears in its own right. E.g. you can't rewrite e^i*pi=-1 with any number. (I think)