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

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Discussion Overview

The discussion revolves around the equation 2^((-2)^x) = x and explores methods for solving it without the use of a calculator. Participants consider various mathematical approaches, including derivatives and logarithmic transformations, while addressing the complexities involved in defining the expression for negative bases raised to powers.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • Some participants propose using derivatives to analyze the equation but encounter issues with negative values inside logarithmic functions.
  • One participant suggests that the equation cannot be solved exactly in a finite number of steps.
  • Another participant emphasizes the need to define (-2)^x, noting that a negative number raised to most irrational powers is not defined in the real number system.
  • It is mentioned that in the complex range, there is a natural definition for negative bases raised to powers.
  • A participant shares that their calculator returns "false" when attempting to solve the equation, indicating challenges in finding a solution.
  • One participant attempts to manipulate the equation using logarithms but expresses uncertainty about how to proceed after reaching x = (-2)^x.

Areas of Agreement / Disagreement

Participants generally agree on the difficulties of solving the equation without a calculator and the complications arising from the definition of (-2)^x. However, there are multiple competing views regarding the feasibility of finding a solution and the implications of using complex numbers.

Contextual Notes

Limitations include the undefined nature of (-2)^x for most irrational powers in the real number system and the unresolved steps in the mathematical manipulations presented.

hangainlover
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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 [itex](-2)^x[/itex]. A negative number to most irrational powers is not defined.
 
HallsofIvy said:
The first thing you will have to do is define [itex](-2)^x[/itex]. 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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