Editing Logarithm (section)

===Existence===
Let {{mvar|b}} be a positive real number not equal to 1 and let {{math|1=''f''(''x'') = {{mvar|b}}<sup>&thinsp;''x''</sup>}}.

It is a standard result in real analysis that any continuous strictly monotonic function is bijective between its domain and range. This fact follows from the [[intermediate value theorem]].<ref name="LangIII.3">{{Citation|last1=Lang|first1=Serge|title=Undergraduate analysis|year=1997|series=[[Undergraduate Texts in Mathematics]]|edition=2nd|location=Berlin, New York|publisher=[[Springer-Verlag]]|doi=10.1007/978-1-4757-2698-5|isbn=978-0-387-94841-6|mr=1476913|author1-link=Serge Lang}}, section III.3</ref> Now, {{mvar|f}} is [[monotonic function|strictly increasing]] (for {{math|''b'' > 1}}), or strictly decreasing (for {{math|0 < {{mvar|b}} < 1}}),<ref name="LangIV.2">{{Harvard citations|last1=Lang|year=1997|nb=yes|loc=section IV.2}}</ref> is continuous, has domain <math>\R</math>, and has range <math>\R_{> 0}</math>. Therefore, {{Mvar|f}} is a bijection from <math>\R</math> to <math>\R_{>0}</math>. In other words, for each positive real number {{Mvar|y}}, there is exactly one real number {{Mvar|x}} such that <math>b^x = y</math>.

We let <math>\log_b\colon\R_{>0}\to\R</math> denote the inverse of {{Mvar|f}}. That is, {{math|log<sub>''b''</sub>&thinsp;''y''}} is the unique real number {{mvar|x}} such that <math>b^x = y</math>. This function is called the base-{{Mvar|b}} ''logarithm function'' or ''logarithmic function'' (or just ''logarithm'').