Logarithms
For the unique real such that is called the logarithm of to the base .
Referenced by (1 direct, 27 transitive)
Direct references:
Transitive (depth 1):
Transitive (depth 2):
Transitive (depth 3):
- Cross-Entropy
- Entropy Rate
- Gibbs' Inequality
- Chain rule for joint entropy
- joint-entropy-is-less-than-or-equal-to-entropy-of-parts
- KL Divergence
- note-25
- note-8
- proof-of-joint-entropy-is-less-than-or-equal-to-entropy-of-parts
- proof-of-theorem-44
- proof-of-theorem-50
- remark-14
- theorem-12
- Weak Asymptotic Equipartition Property
- theorem-44
- theorem-50-intuition
- Noiseless channel transmitting discrete symbols
Transitive (depth 4):
Note
\@{note-2}
The logarithm extends to complex arguments; see the complex logarithm.
TODO: the log of a product is the sum of the logs.
TODO: the log of a quotient is the difference of the logs.
TODO: the log of a power moves the exponent out front.
Referenced by (1 direct)
Direct references:
The function is concave, i.e. is convex.
Proof
\@{proof-of-log-is-concave}
The second derivative of natural is which is always non-positive. By Second Derivative Test for Convexity, is therefore concave.