2^((-2)^x) =x how do you solve for x? without a calculator?

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The equation 2^((-2)^x) = x presents challenges in solving for x without a calculator, particularly due to the undefined nature of negative bases raised to irrational powers. Attempts to use derivatives lead to complications, including negative values within logarithmic functions. The discussion highlights that defining (-2)^x is crucial, as it is not defined for most irrational powers. While some transformations using logarithms are suggested, they ultimately do not yield a straightforward solution. The consensus is that solving this equation exactly in a finite number of steps is not feasible.
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I thouht about derivatives of both sides.
But it leaves me with a negative number inside ln...
I can't think of any way to solve it without a calculator
help?
 
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hangainlover said:
I thouht about derivatives of both sides.
But it leaves me with a negative number inside ln...
I can't think of any way to solve it without a calculator
help?
It can't be done exactly, in a finite number of steps. :smile:
 
hangainlover said:
I thouht about derivatives of both sides.
But it leaves me with a negative number inside ln...
I can't think of any way to solve it without a calculator
help?
The first thing you will have to do is define (-2)^x. A negative number to most irrational powers is not defined.
 
HallsofIvy said:
The first thing you will have to do is define (-2)^x. A negative number to most irrational powers is not defined.

In the complex range however they have a natural definition.
 
I don't think a calculator's going to help much either. My TI-89, with command "solve(2^((-2)^x,x)", returns "false". :)
 
2^((-2)^x)=x
log2((-2)^x)-log2(x)=0
log2(((-2)^x)/x)=0
((-2)^x)/x=1
x=(-2)^x
2^x=(-2)^x=x
I don't know how to proceed.
 

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