Homework Help: Problem on exponent

1. Mar 25, 2013

altamashghazi

1. The problem statement, all variables and given/known data
x^X^x^x^x=2. find value of x.

2. Relevant equations

taking log both sides, but it makes a equation which i am not able to solve.

3. The attempt at a solution
x^(x)^4=2
x^4logx=log2. what next??????

2. Mar 25, 2013

Mentallic

Is that

$$x^{x^{x^{x^x}}}$$

or

$$(((x^x)^x)^x)^x$$

?

3. Mar 25, 2013

altamashghazi

the fist one

4. Mar 25, 2013

altamashghazi

the first one

5. Mar 25, 2013

Mentallic

Was this a problem you just made up? In any case, I don't believe you'll be able to solve it using analytic functions.

If you take the log of both sides, it becomes

$$x^{x^{x^x}}\log x = \log 2$$

and not $x^4\log x$ as you suggested. Also, the power tower isn't

$$x^{x^4}$$ either. If you use Knuth's up arrow notation, then it would be equivalent to x^^5 where each ^ represents an up arrow.

6. Mar 25, 2013

altamashghazi

then what should i do how to solve

7. Mar 25, 2013

Curious3141

Funnily enough, you can solve the infinite power tower quite easily, provided a solution exists, e.g.

$$x^{x^{x^{x^{x}...}}} = 2 \implies x^2 = 2 \implies x = \sqrt{2}$$

But there's no way of solving a finite power tower equation algebraically. You can probably do it numerically, though, to get an approximate solution.

EDIT: you can use $x = \sqrt{2}$ as an excellent starting guess for a numerical solver for your finite power tower problem.

Last edited: Mar 26, 2013
8. Mar 25, 2013

Mentallic

Last edited by a moderator: May 6, 2017
9. Mar 26, 2013

sambarbarian

Curious3141 , I am curious to know how x^x^x .... = 2 became x^2 = 2 .

10. Mar 26, 2013

Mentallic

If x^x^x... = 2 then if we add another x at the bottom of that tower power, we haven't changed anything since the power tower is infinitely high. Or in other words,
y^x^x^x... = 2 where y=x is still equivalent to x^x^x..., so we substitute 2 for x^x^x... since that's what we assumed it is equal to, and arrive at

y^2 = 2
y=x
x^2 = 2

Curious tends to instigate these feelings in all of us

Last edited: Mar 26, 2013