Nash Equilibrium Utlitiy Function: solve ax^(a-2)=1/PI*x for X
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mmm4444bot
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Sure. What have you tried or thought about so far?
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By inspection, one solution is x = 0. I suspect that you'll want to ignore that solution and find the other one.
The other solution can be written in terms of e, by taking the Natural Log of each side. But, you'll first want to isolate a single power of x on one side of the equation. (Right now, x appears on both sides.)
This will require a number of steps, using a few of the Properties of Exponents.
After you take the Natural Log of each side, you can tidy up the result, using some Properties of Logs.
So, to start, rearrange the given equation to a single power of x on one side, with everything else on the other side.
If you need more help, please follow the guidelines and show us what you've done. :cool:
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As written above, the equation is ambiguous. Do you mean either of the following?
. . . . .\(\displaystyle \mbox{1. }\, a\, x^{a-2}\, =\, \frac{1}{\pi}\, x\)
. . . . .\(\displaystyle \mbox{2. }\, a\, x^{a-2}\, =\, \frac{1}{\pi\, x}\)
Thank you!
mmm4444bot
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It could be, if someone is not following the Order of Operations (a common lament). I assumed multiplications and divisions done from left to right, in the order in which they occur, so I didn't see ambiguity. I also noted that they showed an explicit multiplication operator between 1/Pi and x; they didn't between a and x^(a-2).
So, 1/Pi first, then multiply by x.
I pick Door #1.