Is [itex]x = - lo{g_2}(x)[/itex] a complex number

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Discussion Overview

The discussion revolves around the equation x = - \log_2(x) and whether the solutions to this equation must be complex numbers. Participants explore the nature of the solutions, including both real and complex possibilities.

Discussion Character

  • Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant questions if x must be a complex number given the equation.
  • Another participant notes that there are complex solutions, but also mentions the existence of at least one real solution.
  • Some participants discuss the behavior of the left-hand side (LHS) and right-hand side (RHS) of the equation at specific points, indicating that at x = 1, the LHS is larger, and at x = 1/2, the RHS is larger, suggesting a real solution exists between these values.
  • A later reply reiterates the previous point about the existence of a real solution based on the continuity of the functions involved.
  • One participant clarifies that the expression is an equation, not a function.

Areas of Agreement / Disagreement

Participants express differing views on the nature of the solutions, with some asserting the presence of real solutions while acknowledging complex solutions exist. The discussion remains unresolved regarding the necessity of x being a complex number.

Contextual Notes

Participants do not fully resolve the implications of the equation, and there may be missing assumptions regarding the definitions of the logarithmic function and its domain.

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If I have a function such that [itex]x = - lo{g_2}(x)[/itex], then must x be a complex number? Thanks.
 
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There are complex solutions to this, but there is one solution that is real.
 
At x = 1, LHS is the larger; at x = 1/2, RHS is the larger. Since both functions are continuous in that range, there must be a real solution between the two.
 
haruspex said:
At x = 1, LHS is the larger; at x = 1/2, RHS is the larger. Since both functions are continuous in that range, there must be a real solution between the two.
That's the number I was thinking of.
 
also, it is an equation, not a function
 

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