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
