MHB The Limited Logarithm: Why x Can't Be <= 0

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The discussion centers on the limitations of logarithmic functions, specifically why the base x cannot be less than or equal to zero. It explains that the equation P = log_x(Q) translates to Q = x^P, raising concerns about defining expressions like x^{-1} or x^{1/2} when x is zero or negative. To avoid complications and the need for special definitions in these cases, logarithms are restricted to positive bases. This ensures consistency and clarity in mathematical definitions. Consequently, logarithmic functions are only defined for positive bases to maintain their validity.
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Why x can't be less or equal to zero?
 

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roni said:
Why x can't be less or equal to zero?
P= log_x(Q) is equivalent to Q= x^P. How, exactly, would you define x^{-1} if x were equal to 0? How would you define x^{1/2} if x were -1? In order that we not have to give special definitions to cases like those, we only define the exponential for positive base. And from that, logarithm can only be defined for positive base.
 

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