Finding a relationship between functions

[impendingChaos]

I posted on this tangent a little while ago but I've moved forward and was looking for some input.
I have two exponential equations each is described by a different variable (n and v respectively):

y=1.44E-24*exp(46.22n)+2.006E-8
y=2.88*exp(-2.4v)-3.009E-5

Since the end constants are so small I am considering them negligible so:

y=1.44E-24*exp(46.22n)
y=2.88*exp(-2.4v)

What I am trying to find is a relationship between variables n and v.

Currently I've tried taking the natural log of both sides to get:

ln(y)=ln(1.44E-24)+ln(exp(46.22n))
ln(y)=-54.897+56.22n
and
ln(y)=ln(2.88)+ln(exp(-2.4v))
ln(y)=1.058-2.4v

So in essence I now have two linear equations. Are there any suggestions to finding a numerical relationship between n and v. Is there value in finding the intersection of these two linear equations?

Thanks
C.N.

 

[matt grime]

If you don't mind me saying so, there is something odd in you asserting that things of the order 10^-5 are negligible, but that 10^-24 isn't.

y=something
y=something else

therefore something equals something else. That is a relation ship. It just isn't of the (unjustifiable preferred?) v=function of n.

 

[tacman]

Hi C.N.,

Thank you for sharing your progress and asking for input on finding a relationship between your two exponential equations. It looks like you've made some good progress so far by simplifying your equations and taking the natural log of both sides.

One suggestion I have is to plot your two equations on a graph and see if you notice any patterns or similarities. This can help you visually see if there is a relationship between n and v. You can also try plugging in different values for n and v and see if there is a consistent relationship between the resulting y values.

Another approach you could try is to solve for one variable in terms of the other in each equation. For example, in the first equation, you could solve for n by dividing both sides by 1.44E-24 and then taking the natural log, giving you n = (ln(y) + 54.897)/56.22. Then, in the second equation, you could solve for v in terms of n by dividing both sides by 2.88 and taking the natural log, giving you v = (ln(y) - 1.058)/(-2.4). Now, you have both n and v expressed in terms of y, which may help you see a relationship between n and v.

As for the value in finding the intersection of the two linear equations, it could potentially give you a specific value for n and v that satisfies both equations. However, it may not necessarily give you a clear relationship between the two variables.

I hope this helps and good luck with your exploration! Keep experimenting and you may discover a relationship between n and v that you hadn't thought of before.