Real logarithm

Set

context	$x,b\in\mathbb R_+^*$
context	$b\neq 1$
definiendum	$\log_b(x):\mathbb R_+^\to \mathbb R_+^$
definiendum	$\log_b(x):= y$
postulate	$b^y=x$

The logarithm function is that of the Dimension

Consider

$\log_r(r^n/r^1) = \log_r(r^{n-1}) = n - 1 = \log_r(r^n) - \log_r(r^1)$

vs.

${\mathrm {dim}}({\mathbb R}^n/{\mathbb R}^1) = {\mathrm {dim}}({\mathbb R}^{n-1}) = n - 1 = {\mathrm {dim}}({\mathbb R}^n)-{\mathrm {dim}}({\mathbb R}^1)$

where by ${\mathbb R}^n/{\mathbb R}^1$ we mean a quotient vector space .

Theorems

$\log_\mathrm{e}=\mathrm{ln}$

Reference

Wikipedia: Exponentiation

Element of

Context

Complex exponents with positive real bases

Real logarithm

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Theorems

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Real logarithm

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Theorems

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