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Unit of Measure of Exponentiated Item

  1. May 30, 2012 #1
    Let's say we have the quantity
    where x has no unit of measure. What is the unit of measure of f, once we take f^t, where t can be in years?
  2. jcsd
  3. May 30, 2012 #2


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    Welcome to PF!

    Hello Steve! Welcome to PF! :wink:

    Still no units. :smile:

    (ft = etlnf = ∑(tlnf)n/n!)
  4. May 30, 2012 #3
    Hi tiny-tim. Thanks for the reply.
    Just a quick thought, if we say
    f^t=exp[t ln f]
    then we still have that t is years, and exp[time] can't be ok.
    Am I right?
    Thanks for your help
  5. May 30, 2012 #4


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    ah, it would have to be ft/to :wink:
  6. May 30, 2012 #5
    It's not valid to take f^t, with t in years. If f is dimensionless, the exponent has to also be a dimensionless number.
  7. May 31, 2012 #6
    Thanks Khashishi.
    Can I ask, if the exponent is not dimensionless (as tiny-tim suggested above, by saying it should be t/t0) then does it mean that f must have units that I didn't know about or expect?

    Rather, let me ask this: what units would f have, if the exponent has units of time?

    Thanks guys
  8. Jun 1, 2012 #7
    Let's generalize. We have the quantity f. Let's say f is distance, so it is in units of meters.
    Taking f^2 would give square meters.
    Buy let's take it to an exponent that has units, like time.
    f^t is now in what units?
  9. Jun 1, 2012 #8
    the t in that equation should be dimensionless. So, either simply call it the "number" of seconds (or minutes, or years, whatever) or raise f to something like:

    f^(t/[1 sec]) to yield a dimensionless number in the exponent.

    Good thread here:
  10. Jun 1, 2012 #9

  11. Jun 1, 2012 #10
    Usually, you have a time expression like:
    [itex]Y=A \exp(-t/\tau)[/itex]
    where tau is a time constant with the same units as t, so the argument to exp is dimensionless.

    Mathematically, you can absorb the time constant into the base of the exponent since
    [itex]A \exp(-t/\tau) = \exp(1/\tau)^{-t} = f^{-t}[/itex]
    So, f needs to have units of [itex]\exp(1/years)[/itex] to match up with t in years. No one in their right mind would do something like this, but it makes mathematical sense.
  12. Jun 1, 2012 #11


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    There's no reason to expect that you can use a quantity as an exponent. After all, you only need to say, in words, what "the exponent" means. It means the number of times that a number is multiplied by itself and it would be daft to say "Mutiply 3 by itself five point three inches times". Go back to basics for the answers to this sort of question.
  13. Jun 1, 2012 #12
    This is only one definition, and rather elementary and limited. A lot of natural phenomena exhibit exponential growth or decay. It's probably better to view an exponential as a function whose derivative is proportional to itself.
  14. Jun 1, 2012 #13


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    I disagree entirely (and most humbly:wink:). Exponential growth is exactly what happens when a fractional increase is repeated a number of times.
    Your more sophisticated version is very useful but it's only describing a consequence of the process.
    Last edited: Jun 1, 2012
  15. Jun 1, 2012 #14
    the best way to view exponential (natural) growth/decay is to say 'rate of change is proportional to how much you have got'.....this is where I start with students and they seem to be able to relate it to money and savings and interest rates as well as physical phenomena such as radioactive decay
    i.e dA/dt = +/-constant x A

    This is exactly the same as saying that you get the same fractional increase or decrease
    per unit time.
  16. Jun 1, 2012 #15
    The exponent does not have units/dimensions.
    It is the powerthat a number (e) is raised to.... just a number.
    In the same way a log has no units/dimensions.... it is just a number
  17. Jun 1, 2012 #16


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    But your conceptual definition only makes sense for integer exponents. You have to introduce more advanced concepts like the idea of a limit to deal with the more general case anyway, am I right?
  18. Jun 2, 2012 #17


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    Mine is a simple, starting definition, true, but it extends, without too much imagination, to non-integers. And, as far as the original question goes, it establishes a logical reason why the index is dimensionless. The logic doesn't change.
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