Editing Logarithm (section)

===Number theory===
[[Natural logarithm]]s are closely linked to [[prime-counting function|counting prime numbers]] (2, 3, 5, 7, 11, ...), an important topic in [[number theory]]. For any [[integer]]&nbsp;{{mvar|x}}, the quantity of [[prime number]]s less than or equal to {{mvar|x}} is denoted {{math|[[prime-counting function|{{pi}}(''x'')]]}}. The [[prime number theorem]] asserts that {{math|{{pi}}(''x'')}} is approximately given by
<math display="block">\frac{x}{\ln(x)},</math>
in the sense that the ratio of {{math|{{pi}}(''x'')}} and that fraction approaches 1 when {{mvar|x}} tends to infinity.<ref>{{Citation|last1=Bateman|first1=P.T.|last2=Diamond|first2=Harold G.|title=Analytic number theory: an introductory course|publisher=[[World Scientific]]|location=New Jersey|isbn=978-981-256-080-3 |oclc=492669517|year=2004}}, theorem 4.1</ref> As a consequence, the probability that a randomly chosen number between 1 and {{mvar|x}} is prime is inversely [[proportionality (mathematics)|proportional]] to the number of decimal digits of {{mvar|x}}. A far better estimate of {{math|{{pi}}(''x'')}} is given by the [[logarithmic integral function|offset logarithmic integral]] function {{math|Li(''x'')}}, defined by
<math display="block"> \mathrm{Li}(x) = \int_2^x \frac1{\ln(t)} \,dt.  </math>
The [[Riemann hypothesis]], one of the oldest open mathematical [[conjecture]]s, can be stated in terms of comparing {{math|{{pi}}(''x'')}} and {{math|Li(''x'')}}.<ref>{{Harvard citations|last1=Bateman|first1=P. T.|last2=Diamond|year=2004|nb=yes |loc=Theorem 8.15}}</ref> The [[Erdős–Kac theorem]] describing the number of distinct [[prime factor]]s also involves the [[natural logarithm]].

The logarithm of ''n'' [[factorial]], {{math|1=''n''! = 1 · 2 · ... · ''n''}}, is given by
<math display="block"> \ln (n!) = \ln (1) + \ln (2) + \cdots + \ln (n).</math>
This can be used to obtain [[Stirling's formula]], an approximation of {{math|''n''!}} for large {{mvar|n}}.<ref>{{Citation|last1=Slomson|first1=Alan B.|title=An introduction to combinatorics|publisher=[[CRC Press]]|location=London|isbn=978-0-412-35370-3|year=1991}}, chapter 4</ref>