A bit of help needed solving x^x

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margeman2k3
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A bit of help needed solving x^x

Postby margeman2k3 » Fri Mar 07, 2008 9:13 am UTC

Okay, I was looking over some notes, and I started to wonder how you would solve an equation like xx = 12.
Now, I know that x is on (2,3), but that's about it.

I know that usually you would take logx of both sides which would isolate the exponent, or even take the xth root of both sides to isolate the base.

I've tried both methods, but nothing seems to work. Whenever I do anything with logs and radicals, I end up with ln(xx) = ln(12), which just takes me back to the original equation.

Can anyone shed some light on this for me?

In case you're wondering where I got this, I was looking at nominal interest rates compounded n times per year.
Essentially, you would have FV = (1 + i(n)/n)n
If you tried to solve for n (without a calculator) you would have a similar problem (as far as i can tell) [[see spoiler]]

Spoiler:
Let FV = 1000 and i(n) = 10% = 0.1
Then, 1000 = (1 + 0.1/n)n
(1000)1/n = 1 + 0.1/n
1000n = n / (n + 0.1)

And from there, I seem to keep going in circles.
1000n (n + 0.1) = n
1000n (1 + 0.1/n) = 1
1 + 0.1/n = 1000-n
1000 = (1 + 0.1/n)n


Also, I started thinking that you could use linear approximation for this. Can anyone confirm/deny that?

Thanks for your help!

Robin S
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Re: A bit of help needed solving x^x

Postby Robin S » Fri Mar 07, 2008 9:24 am UTC

I believe problems like yours have been posted on these forums before. You want the Lambert W function. Check out example #2 from that article.
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gmalivuk
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Re: A bit of help needed solving x^x

Postby gmalivuk » Fri Mar 07, 2008 5:08 pm UTC

I'm starting to think a sticky thread all about that function would be a good idea, but I don't know what I'd call it to make sure people with these sorts of questions read it before posting...
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Re: A bit of help needed solving x^x

Postby Ambush Commander » Sat Mar 08, 2008 5:31 am UTC

Well, I think this is a legitimate question, because it's somewhat difficult to Google "x^x". It's a three word answer, and the reader can go off and read about the problem, and you are only out perhaps ten seconds of your time.
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gmalivuk
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Re: A bit of help needed solving x^x

Postby gmalivuk » Sat Mar 08, 2008 5:37 am UTC

I know it's a legitimate question. But it's one we get asked a lot, and the answer is always those same three words. So it would be nice if there were some simple way to make a sticky thread about it.
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