Intro to Logarithms (article) | Logarithms | Khan Academy

Logarithms are undefined for base 1 because there exist no real power that we could raise one to that would give us a number other than 1. In other words:
1ˣ = 1
For all real 𝑥. We can never have 1ˣ = 2 or 1ˣ = 938 or 1ˣ = any number besides 1.
If the base of the logarithm is negative, then the function is not continuous. For instance, sure the logarithm is defined for even and odd powers of negative numbers (though even powers are positive and the odd powers a negative and this is a wild jumping behavior that will continue for all integers). However, what about values between the integers? For instance, what if I asked you what power I needed to raise -2 to in order to get 1/2? The answer is a complex number, and it can only be found with some knowledge of trigonometry and the de’Moivre’s theorem. In other words, there are gaps between the integer powers where the function is only defined in the nonreal numbers. The only places where it is defined (in the real numbers) is for integer powers, and plotting just those clearly don’t give a continuous curve.

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