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

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The equation x = -log₂(x) can have both complex and real solutions. A real solution exists between x = 1 and x = 1/2, as the left-hand side (LHS) is greater at x = 1 and the right-hand side (RHS) is greater at x = 1/2. The continuity of both sides in this range guarantees at least one real solution. Additionally, the discussion clarifies that the expression is an equation, not a function. Therefore, while complex solutions exist, there is a valid real solution as well.
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If I have a function such that x = - lo{g_2}(x), 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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