Unit of Measure of Exponentiated Item

[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

 

[tiny-tim]

Welcome to PF!

Hello Steve! Welcome to PF!

Still no units.

(f^t = e^tlnf = ∑(tlnf)ⁿ/n!)

 

[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

 

[tiny-tim]

ah, it would have to be f^t/t_o

 

[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.

 

[Steve Zissou]

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.

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

 

[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?

 

[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:

 

[Steve Zissou]

Travis_King

Thanks

 

[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.

 

[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.

 

[Khashishi]

It means the number of times that a number is multiplied by itself

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.

 

[sophiecentaur]

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.

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.

 

[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.

 

[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

 

[cepheid]

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.

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?

 

[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.