Editing Logarithm (section)

===Entropy and chaos===
[[File:Chaotic Bunimovich stadium.svg|right|thumb|[[Dynamical billiards|Billiards]] on an oval [[billiard table]]. Two particles, starting at the center with an angle differing by one degree, take paths that diverge chaotically because of [[reflection (physics)|reflections]] at the boundary.|alt=An oval shape with the trajectories of two particles.]]

[[Entropy]] is broadly a measure of the disorder of some system. In [[statistical thermodynamics]], the entropy&nbsp;{{mvar|S}} of some physical system is defined as
<math display="block"> S = - k \sum_i p_i \ln(p_i).\, </math>
The sum is over all possible states&nbsp;{{Mvar|i}} of the system in question, such as the positions of gas particles in a container. Moreover, {{math|''p''<sub>''i''</sub>}} is the probability that the state&nbsp;{{Mvar|i}} is attained and {{mvar|k}} is the [[Boltzmann constant]]. Similarly, [[entropy (information theory)|entropy in information theory]] measures the quantity of information. If a message recipient may expect any one of {{mvar|N}} possible messages with equal likelihood, then the amount of information conveyed by any one such message is quantified as {{math|log<sub>2</sub>&thinsp;''N''}} bits.<ref>{{Citation|last1=Eco|first1=Umberto|author1-link=Umberto Eco|title=The open work  |publisher=[[Harvard University Press]]|isbn=978-0-674-63976-8|year=1989}}, section III.I</ref>

[[Lyapunov exponent]]s use logarithms to gauge the degree of chaoticity of a [[dynamical system]]. For example, for a particle moving on an oval billiard table, even small changes of the initial conditions result in very different paths of the particle. Such systems are [[chaos theory|chaotic]] in a [[Deterministic system|deterministic]] way, because small measurement errors of the initial state predictably lead to largely different final states.<ref>{{Citation | last1=Sprott | first1=Julien Clinton | title=Elegant Chaos: Algebraically Simple Chaotic Flows | journal=Elegant Chaos: Algebraically Simple Chaotic Flows. Edited by Sprott Julien Clinton. Published by World Scientific Publishing Co. Pte. Ltd | url={{google books |plainurl=y |id=buILBDre9S4C}} | publisher=[[World Scientific]] |location=New Jersey|isbn=978-981-283-881-0| year=2010| bibcode=2010ecas.book.....S | doi=10.1142/7183 }}, section 1.9</ref> At least one Lyapunov exponent of a deterministically chaotic system is positive.