Question

For the equation m(a^r) + v = n(b^r), solve for r. All variables other than r are constants. My trouble is I'm finding it difficult to isolate r... is there a relatively trivial solution? I keep thinking logarithms, but I'm getting nowhere...

Answer 1

u can try differentiation
d(a^x)/dx=a^x * ln a

Answer 2

@ambius

Answer 3

try differentiation. It will eliminate v.

Answer 4

on differntiating with r
m*a^r*lna +0 = n*b^r*lnb

Answer 5

lol, it's funny... I had that though earlier... but I didn't think the problem should require any machinery as heavy as differentiation given the expectations of the class I'm taking... but I guess it may be the easiest means to solve it.

Answer 6

a^r/b^r =n*lnb/(m*ln a)
(a/b)^r=(n/m)*(lnb/lna)

Answer 7

now take log both side

Answer 8

r*ln(a/b)=ln((n/m)*(lnb/lna))

Answer 9

@ambius

Answer 10

So, what is your question basically ?

Answer 11

I think its been answered, thanks. I guess short of some kind of numerical method, simple differentiation is the best solution.

Answer 12

or divide both sides by v and take log

Answer 13

The last suggestion doesn't work very well. v/v = 1 ... which is still a constant. Taking the log of ma^r + v is not easily solved with logarithms.

Answer 14

sorry buddy

Answer 15

lol, no problem. I've had quite a bit of time to think about the problem. I tried re-working the original equation to get rid of the +v... but any other methods to derive the appropriate equation including the r still results in the addition of a constant.

Answer 16

I'm just surprised because I don't think our teacher realized the problem isn't solvable without something like calculus.

Answer 17

Not that it's important as the question has been answered, but to give you the idea of the actual question, it involves sequences.

The population of birds on island A increases at 5% per year, and 200 birds migrate to island B every year.

The population of birds on island B increases at 3% per year, and because of the consequences of migration from island A, it's population also increases by 200 birds every year.

Given the initial conditions that the population of island A = 5000 and B=4500, determine when the populations are equal.

Answer 18

Created a sequence to describe the yearly population growth, then turned them into functions. I then set those functions equal to each other, and the resultant equation was in the general form of the question I posted, lol

Answer 19

That's just the background in case anyone is board enough to want to take a crack at the problem. I'd be interested to know if there is a simpler solution not requiring calculus or approximation/numerical methods.

Answer 20

1) Differentiation is kind of magic on this one. A little unusual, but if it works, it works. Differentiation normally can be used to help narrow down the location of solutions. It can also help as an integral part of a numerical solution.

2) It is possible, if there were more information about m, a, n, and b, that there would be an analytic solution. As it stands, it cannot be done in a general sense.

3) It is a common fallacy to think of numerical methods as "approximate".

I do not believe this problem statement is asking for a general solution. I also do not believe you have formulated it correctly. If you intend only one migration, you do not need the exponent, r. If you intend multiple migrations, that v needs a multiple. Also, the migrated birds get to reproduce, don't they?

I would suggest \(m\cdot a^{r} - v\cdot\dfrac{1-a^{r}}{1-a}\) for the annual population of Island A and \(n\cdot b^{r} + v\cdot\dfrac{1-b^{r}}{1-b}\), where a = 1.05 and b = 1.03 in your initial conditions.

Island A
A0 = 5000
A1 = 5000*1.05 - 200 = 5050
A2 = 5050*1.05 - 200 = 5102
A3 = 5102*1.05 - 200 = 5157

Island B
B0 = 4500
B1 = 4500*1.03 + 200 = 4835
B2 = 4835*1.03 + 200 = 5180
B3 = 5180*1.03 + 200 = 5535

Pretty obviously, it is less than 2 years.

A1.5 = 5050*1.05^(1/2) - 100 = 5074
B1.5 = 4835*1.03^(1/2) + 100 = 5006

A1.75 = 5050*1.05^(3/4) - 150 = 5088
B1.75 = 4835*1.03^(3/4) + 150 = 5093

Pretty close. We've started to chop up birds in order to make the continuous model match the discrete reality. We're not going to get much closer than that.

Anyway, some random musings, since you left it open.

Answer 21

Lol, you're analysis is quite correct. I got the exact equations you mentioned. I would have felt that your answer would have been suffice in any real-world context. However, the teacher wanted us to find the percent error of a visual approximation (based on the plotted data) after finding the exact solution for 'r' algebraically.

But thanks for your musings - it only further verifies my thought processes. Again, it's just interesting as I believe the question may be more involved than was originally intended - I expected (and probably, so does my class) that the answer shouldn't have needed anything more complicated than algebra or logarithms.

Thanks everyone for your help, thoughts, and time. :)