Can You Solve n to the Power of n = 240?

[Alan A]

Please can you help on this one?

 

[Phrak]

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

 

[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 don't know the number we can't know the log.
Greetings
Alan

 

[Borek]

If we don't know the number we can't know the log.

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

 

[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

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

 

[Redbelly98]

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.

 

[uart]

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

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.

 

[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,

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

 

[ehild]

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 don't know the number we can't know the log.
Greetings
Alan

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