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I know this question has been asked before, but I am still now convinced with the explanations from previous questions like this one:

Why must the base of a logarithm be a positive real number not equal to 1?

Say a function $f$ follows the rule $f: D \to \mathbb R, f(x) = \log_{-2}{x}, D \in \mathbb Z$

The real range is definitely $\mathbb Z^+$.

However, the real domain, $D$ can be defined as in two parts where all the odd powers of $-2$ result in $-x$: ${(-2, -8, -32...)}$ and the even powers will similarly be $+x$.

What I do want to know is why do logarithmic functions with negative bases not exist (undefined) when their inverse exist for negative values of $x$ exist and therefore why isn't $D$ defined for certain, negative and positive set of values, as the function theoretically exists?

Dstarred
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1 Answers1

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$$\text{Using change of base}, \\\log_{-2} x=\frac{\log x}{\log (-2)}$$

$$\text{But} \log (-2)=y\Rightarrow10^y=-2\text{ which is impossible for real values $y$.}$$

F. A. Mala
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  • I know it does not work when using change of base laws, but does that mean we just ignore other ways to solve it (that work)?? – Dstarred Sep 23 '21 at 10:08
  • No doubt, you may choose some appropriate domain and codomain of the $\log$ function and make it work. I doubt it won't be a "good-looking" domain/codomain. – F. A. Mala Sep 23 '21 at 10:15