===== Real logarithm =====
==== Set ====
| @#55CCEE: context     | @#55CCEE: $ x,b\in\mathbb R_+^* $ |
| @#55CCEE: context     | @#55CCEE: $ b\neq 1 $ |
| @#FFBB00: definiendum | @#FFBB00: $\log_b(x):\mathbb R_+^*\to \mathbb R_+^*$ |
| @#FFBB00: definiendum | @#FFBB00: $\log_b(x):= y$ |
| @#55EE55: postulate   | @#55EE55: $b^y=x$ |

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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: [[http://en.wikipedia.org/wiki/Exponentiation|Exponentiation]]

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=== Element of ===
=== Context ===
[[Complex exponents with positive real bases]]