i need help with a challenging math problem plz.?

Question

x^x^x^...=2
asumming that this is possible.

(it means x to the x to the x to infinity equals 2)

i think x will have to include sq. rt. of 2 and infinity or something. However i know that it isnt sq. rt. of 2 to infinity power. I think...

plz show work. Thanks.

Answer 1

exponentiation is not commutative so you have to be specific as to what you mean. Do you mean
((x^x)^x)^x...
or do you mean
(x^(x^(x^(x...

The second one is easy to solve. Notice that
(x^(x^(x^(x...
contains itself. I.e., it equals
x ^ (the original sequence)
because the sequence is infinite.

So, 2 = (x^(x^(x^(x... = x^(2) = x squared.
So if x squared = 2. then x = square root of 2.

Answer 2

x = (2)^1/2^1/2^1/2^1/2 . . . . .

Answer 3

As x --> 0+, f(x) --> 1
As x --> 0 - f(x) --> -1
if x>1 f(x) --> infinity
Looks like it's impossible

Answer 4

If x^{x^(x^...)} = 2, then the term in curly brackets, which is x^(x^...), is also 2; so

x^2 = 2

Therefore, x = sqrt(2)