Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Question about the exponential growth and decay formula

  1. Jul 14, 2014 #1
    [I don't know if this is in the right topic or not so I hope I'm all good]

    My question is related to the exponential growth and decay formula Q=Ae^(kt).

    Simply, why is the value e used as the base for the exponent?

    Does it have to be e?
    If so, can anybody tell me why? Thanks


    ~| FilupSmith |~
     
    Last edited: Jul 14, 2014
  2. jcsd
  3. Jul 14, 2014 #2

    olivermsun

    User Avatar
    Science Advisor

    It doesn't have to be e — it could actually be any "base."

    For example, check out the first form for exponential decay given here; the base is 2 (or 1/2) because here the exponent is scaled by the half-life.

    You can convert between different bases; here 2 is easy because it makes sense in this particular problem, other times e is used because it's convenient for doing further math like taking derivatives and integrals.
     
  4. Jul 14, 2014 #3
    I thought so, I initially noticed this when solving for t. Because e kind of disappears once you take the natural log. I guess its really for convenience?

    None the less, thank you very much!


    ~| FilupSmith |~
     
  5. Jul 14, 2014 #4

    Matterwave

    User Avatar
    Science Advisor
    Gold Member

    Notice that: $$x^{kt}=e^{kt\ln x}$$

    So I can convert from any base x to ##e## by multiplying the constant ##k## by ##\ln x##. So it's not mandatory that we use base e. However, base e is nice basically because of the integral:

    $$\int \frac{1}{x}dx=\ln(x)+C$$

    When solving the differential equation, it's sort of natural to stay in base e.
     
  6. Jul 14, 2014 #5
    That helps a lot! Thanks!


    ~| FilupSmith |~
     
  7. Jul 14, 2014 #6

    Erland

    User Avatar
    Science Advisor

    No, it doesn't need to be ##e##. One can use any positive number except ##1## as base. In general, if ##a>0##, ##a \neq 1##: ##e^{kt}=a^{(k/\ln a)t}##. In particular, if ##a=e^k##, we get the simple form ##e^{kt}=a^t##.

    The reason that the base ##e## is often used is that it simplifies differentiation: The derivative of ##e^{kt}## is ##ke^{kt}##, while the derivative of ##a^{kt}## is ##k\ln a\, a^{kt}##.

    This means that ##y=Ce^{kt}## is the general solution of the differential equation ##dy/dt=ky##, which arises naturally in the study of growth and decay. With base ##a##, the solution gets the more complicated form ##y=Ca^{(k/\ln a)t}##.

    So, the base ##e## a simplifies the calculations.
     
  8. Jul 14, 2014 #7

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    However, [itex]a^x= e^{ln(a^x)}= e^{x ln(a)}[/itex]. In other words, it is fairly easy to convert an exponential to any base to an exponential to any other base. We typically use "e" because it is simplest- it has the property that its derivative is just itself while the derivative of any other base is a constant, other than 1, times the function.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Question about the exponential growth and decay formula
Loading...