X's and 3

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Fatin Farhan
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X's and 3

Unread post by Fatin Farhan » Wed Sep 03, 2014 1:47 pm

what is the largest real value of $$x$$ such that
$$x^{x^{x^{x^{\cdots x^{x^3}}}}}=3$$
Hint
$$f(x)=x \times ln (x)$$ is negative for $$x<1$$
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Nirjhor
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Re: X's and 3

Unread post by Nirjhor » Wed Sep 03, 2014 3:15 pm

How many \(x\)'s are here?
- What is the value of the contour integral around Western Europe?

- Zero.

- Why?

- Because all the poles are in Eastern Europe.


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Thanic Nur Samin
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Re: X's and 3

Unread post by Thanic Nur Samin » Fri Sep 05, 2014 10:47 am

No matter how many x's are there, the answer would be $\sqrt[3]{3}$
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Fatin Farhan
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Re: X's and 3

Unread post by Fatin Farhan » Fri Sep 05, 2014 4:06 pm

Here we have finitely many $$x's$$. But where's the prove
No matter how many x's are there, the answer would be $$\sqrt[3]{3}$$
:?
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Thanic Nur Samin
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Re: X's and 3

Unread post by Thanic Nur Samin » Fri Sep 05, 2014 6:23 pm

If x=$\sqrt[3]{3}$,then $x^3=3$.

Let $\sqrt[3]{3}^{\sqrt[3]{3}^{\sqrt[3]{3}^{\sqrt[3]{3}^{\cdots \sqrt[3]{3}^{\sqrt[3]{3}^3}}}}}=3$
We know that $3=3$
$\rightarrow \sqrt[3]{3}^3=3$
Since $3=\sqrt[3]{3}^{\sqrt[3]{3}^{\sqrt[3]{3}^{\sqrt[3]{3}^{\cdots \sqrt[3]{3}^{\sqrt[3]{3}^3}}}}}$, We replace and find,
$\sqrt[3]{3}^{(\sqrt[3]{3}^{\sqrt[3]{3}^{\sqrt[3]{3}^{\sqrt[3]{3}^{\cdots \sqrt[3]{3}^{\sqrt[3]{3}^3)}}}}}}=3$

Proved by induction.
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*Mahi*
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Re: X's and 3

Unread post by *Mahi* » Fri Sep 05, 2014 8:00 pm

Now, prove that $\sqrt[3]3$ is the unique solution (or the solution isn't complete).
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Thanic Nur Samin
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Re: X's and 3

Unread post by Thanic Nur Samin » Sat Sep 06, 2014 6:02 pm

If $x<0$, then,
Case 1:x is an integer.
Subcase 1:x is even.
If x is even, then after 3 steps it would go imaginary.
Subcase 2:x is odd.
If x is odd, then $x^{x^{x^{x^{\cdots x^{x^{3}}}}}}$ would be negative.
Case 2:x is a fraction.
If x is a fraction, then $x^{x^{x^{x^{\cdots x^{x^{3}}}}}}$ would be negative.

If $x=0$, then,
$x^{x^{x^{x^{\cdots x^{x^{3}}}}}}$ will be indeterminable.

If $0<x<1$, then,
0<$x^{x^{x^{x^{\cdots x^{x^{3}}}}}}$<1.

If $x=1$, then,
$x^{x^{x^{x^{\cdots x^{x^{3}}}}}}$=1.

If $x>1$, then,
$x^{x^{x^{x^{\cdots x^{x^{3}}}}}}$ is a stictly increasing function, and thus will intersect y=3 only once, at $\sqrt[3]{3}$.
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Re: X's and 3

Unread post by *Mahi* » Mon Sep 08, 2014 7:46 pm

Thanic Nur Samin wrote:If x is even, then after 3 steps it would go imaginary.
What?
Thanic Nur Samin wrote:If x is odd, then $x^{x^{x^{x^{\cdots x^{x^{3}}}}}}$ would be negative.
What?
Thanic Nur Samin wrote:If x is a fraction, then $x^{x^{x^{x^{\cdots x^{x^{3}}}}}}$ would be negative.
What?
Thanic Nur Samin wrote:If $x>1$, then,
$x^{x^{x^{x^{\cdots x^{x^{3}}}}}}$ is a stictly increasing function
Proof?

For the first three cases, are you missing some log somewhere? Because $\text{positive}^{\text{anything}} > 0$ and is real.
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Thanic Nur Samin
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Re: X's and 3

Unread post by Thanic Nur Samin » Wed Sep 10, 2014 6:41 pm

*Mahi* wrote:
Thanic Nur Samin wrote:If x is even, then after 3 steps it would go imaginary.
What?
Thanic Nur Samin wrote:If x is odd, then $x^{x^{x^{x^{\cdots x^{x^{3}}}}}}$ would be negative.
What?
Thanic Nur Samin wrote:If x is a fraction, then $x^{x^{x^{x^{\cdots x^{x^{3}}}}}}$ would be negative.
What?

For the first three cases, are you missing some log somewhere? Because $\text{positive}^{\text{anything}} > 0$ and is real.
Note what is written at the top of all those 3 cases.
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