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

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In summary: General Math". There are other members who would like to help with this problem.In summary, Alan provides a summary of the content and asks that people go back to General Math if they want to help.
  • #1
Alan A
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Please can you help on this one?
 
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  • #2
First, what is the log of n^n in the base n?
 
  • #3
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
 
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  • #4
Alan A said:
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.
 
  • #5
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

[tex]log_{a}X = \frac{log_{b}X}{log_{b}a}[/tex]
 
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  • #6
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
Borek said:
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 [itex]log_n(240)[/itex], 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.
 
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  • #8
Looking at your problem I think that you need to use:

[tex] y=xe^x \iff W(y)=x [/tex]

Your challenge will be to get [tex]n^n=240[/tex] in the form [tex]y=xe^x[/tex]

Notice first that

[tex] 1 = \frac{1}{n}a^\frac{1}{n} [/tex]

and recall,

[tex] a=e^{\ln a} [/tex]
 
  • #9
Alan A said:
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
 

1. What is the value of n in "n to power n = 240"?

The value of n is approximately 4.483, as calculated using a mathematical equation or by trial and error.

2. How do you solve for n in "n to power n = 240"?

To solve for n, you can use a logarithmic function or manipulate the equation to get n on one side and the constant 240 on the other side. Then, use a calculator or trial and error to find the approximate value of n.

3. Can there be more than one solution for n in "n to power n = 240"?

Yes, there can be more than one solution for n. In this case, n can have two solutions - a positive and a negative value. However, for most exponential equations, there is usually only one unique solution.

4. What is the significance of the "n to power n" format in this equation?

The "n to power n" format indicates an exponential equation, where the variable (n) is raised to a power (n). This type of equation is commonly used in mathematics and science to model growth or decay.

5. Is there a general method for solving exponential equations like "n to power n = 240"?

Yes, there are various methods for solving exponential equations, such as using logarithms, manipulating the equation, or using a graphing calculator. The most suitable method may vary depending on the specific equation and its complexity.

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