If $a^x=y$, we can write $x=\log_a y$, what can we write in logarithmic form for $1^5=1$?
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2$a\in R^+ - {1}$ – Lion Heart Oct 10 '22 at 05:09
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Related: https://math.stackexchange.com/questions/413713/log-base-1-of-1 – Oct 10 '22 at 05:17
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The notation $\log_a x$ is well-defined for real numbers $a > 0$ and $x > 0$, except for $a = 1$. If $a = 1$ then $\log_1 x$ is not well-defined for any $x$ because if $x \neq 1$ then there are no solutions to $1^y = x$, and if $x = 1$ then there are infinitely many solutions.
You will notice this also relates to the change-of-base formula for logarithms:
$\log_a x = \frac{\log_b x}{\log_b a}$
If $a = 1$, then $\log_b a = 0$ which makes the right-hand side undefined.
ConMan
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