Solve n to power n = 240

1. Apr 11, 2010

Alan A

Please can you help on this one?

2. Apr 11, 2010

Phrak

First, what is the log of n^n in the base n?

3. Apr 11, 2010

Alan A

Thank you for your interest. In the problem n is an unknown number. So it is a number raised to its own power. It must be less than 4 as 4 to power 4 = 256. If we dont know the number we cant know the log.
Greetings
Alan

Last edited: Apr 11, 2010
4. Apr 11, 2010

Staff: Mentor

Phrak asked about log base n, that one we know. Take a look at log definition.

5. Apr 11, 2010

Phrak

I hope you can bail me out Borek. I'm out on a limb. Tell me never to do homework help again!

Alan, the idea on this forum, as you may know, is to help lead you to the answer rather than giving you the answer. Anyway, the result is going to be a real number not an integer.

3^3= 27, so the answer lies between 3 and 4.

The formula to change between bases is

$$log_{a}X = \frac{log_{b}X}{log_{b}a}$$

Last edited: Apr 11, 2010
6. Apr 11, 2010

Redbelly98

Staff Emeritus
Moderator's note: thread moved from "General Math"

Please do not help further until the OP, Alan A, provides his thoughts on solving the problem.

7. Apr 11, 2010

uart

Maybe the OP was referring to not being able to evalutate the RHS, that is $log_n(240)$, without knowing "n".

To the OP. The answer can not be solved in terms of elementary functions, so just use trial and error or numerical methods for an approximate answer. The answer can however be easily expressed in terms of a special function called the "Lambert W" function, as per my previous (now apparently deleted) post.

Last edited: Apr 11, 2010
8. Apr 11, 2010

Gregg

Looking at your problem I think that you need to use:

$$y=xe^x \iff W(y)=x$$

Your challenge will be to get $$n^n=240$$ in the form $$y=xe^x$$

Notice first that

$$1 = \frac{1}{n}a^\frac{1}{n}$$

and recall,

$$a=e^{\ln a}$$

9. Apr 11, 2010

ehild

Alan, you are quite close to the solution. It is close to 4, but a bit less. Try n= 3.9, 3.95, 3.97 and so on.

ehild