If log 1 to the base 1 can be any number. So we can invent a new mathematical idea called the "any". If log 1(1) = "that idea", so is that wrong?
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There is not a number which "can be any number". – coffeemath Nov 24 '20 at 14:44
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5Does this answer your question? log base 1 of 1 – CPCH Nov 24 '20 at 14:45
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$\log_1 1$ is undefined. By the change of base formula, we can see this is equal to $\frac {\ln 1} {\ln 1} = \frac 0 0$, which is itself undefined (an indeterminate form).
Undefined or indeterminate does not mean "any number", it simply means that it is not possible to assign a proper meaning to the expression.
Deepak
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1Completely agree that "undefined" is not equivalent to "any number". The confusion might arise from the fact that there is not one number that solves the equation - but this doesn't suggest that all numerical solutions are equally valid ("any" number), but rather that all numerical solutions are equally invalid ("not a number"/undefined). Log1(1) doesn't equal 1, or 2, or -73, it equals none of those. – Nuclear Hoagie Nov 24 '20 at 15:02