Solving (1 + 0.2/x)^x = 1.21 for x

[qqpenguin12]

Homework Statement

I hope everyone is having a great day. The equation I am working with at the moment is:

(1+ (.2/x))^(x) = 1.21.

1.21 represents the final value, whereas 1 is the present value. I am trying to solve for x, which represents the number of periods necessary for the entity to grow from 1 to 1.21. I am a little stuck on the algebraic manipulations. If anyone could help that would be awesome, thanks!

Homework Equations

The Attempt at a Solution

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I have attempted rooting the entire expression, as well as using the "e" and "ln" functions, but I am still having trouble isolating x.

 

[mfb]

There is no proper analytic solution to this type of equation.
You can get numeric approximations to arbitrary precision.

 

[qqpenguin12]

There is no proper analytic solution to this type of equation.
You can get numeric approximations to arbitrary precision.

Thank you for clarifying that, as I was pretty confused! Could you tell me where I could find my information on how to do that? Thanks again.

 

[mfb]

"that" means getting numerical approximations? There are many methods, Wikipedia has a long article as usual. Newton's method is the easiest one that works reasonably well in most cases.

If you are just interested in a solution for this specific equation, you can guess some numbers, or use WolframAlpha to find solutions. In this case, the integer solution is exact.

 

[SammyS]

... In this case, the integer solution is exact.

@qqpenguin12,

Notice that (1.1)² = 1.21