Unit of Measure of Exponentiated Item

1. May 30, 2012

Steve Zissou

Hello.
Let's say we have the quantity
f=1/(1+x)
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?
Thanks

2. May 30, 2012

tiny-tim

Welcome to PF!

Hello Steve! Welcome to PF!

Still no units.

(ft = etlnf = ∑(tlnf)n/n!)

3. May 30, 2012

Steve Zissou

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

4. May 30, 2012

tiny-tim

ah, it would have to be ft/to

5. May 30, 2012

Khashishi

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.

6. May 31, 2012

Steve Zissou

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

7. Jun 1, 2012

Steve Zissou

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?

8. Jun 1, 2012

Travis_King

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:

9. Jun 1, 2012

Steve Zissou

Travis_King

Thanks

10. Jun 1, 2012

Khashishi

Usually, you have a time expression like:
$Y=A \exp(-t/\tau)$
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
$A \exp(-t/\tau) = \exp(1/\tau)^{-t} = f^{-t}$
$f=\exp(1/\tau)$
So, f needs to have units of $\exp(1/years)$ to match up with t in years. No one in their right mind would do something like this, but it makes mathematical sense.

11. Jun 1, 2012

sophiecentaur

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.

12. Jun 1, 2012

Khashishi

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.

13. Jun 1, 2012

sophiecentaur

I disagree entirely (and most humbly). 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
14. Jun 1, 2012

truesearch

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.

15. Jun 1, 2012

truesearch

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

16. Jun 1, 2012

cepheid

Staff Emeritus
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?

17. Jun 2, 2012

sophiecentaur

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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