Editing Logarithm (section)

==Analytic properties==
A deeper study of logarithms requires the concept of a ''[[function (mathematics)|function]]''. A function is a rule that, given one number, produces another number.<ref>{{citation | last1=Devlin | first1=Keith | author1-link=Keith Devlin | title=Sets, functions, and logic: an introduction to abstract mathematics | publisher=Chapman & Hall/CRC | location=Boca Raton, Fla | edition=3rd | series=Chapman & Hall/CRC mathematics | isbn=978-1-58488-449-1 | year=2004 | url={{google books |plainurl=y |id=uQHF7bcm4k4C}}}}, or see the references in [[function (mathematics)|function]]</ref> An example is the function producing the {{mvar|x}}-th power of {{mvar|b}} from any real number&nbsp;{{mvar|x}}, where the base&nbsp;{{mvar|b}} is a fixed number. This function is written as {{math|1=''f''(''x'') = {{mvar|b}}<sup>&thinsp;''x''</sup>}}. When {{mvar|b}} is positive and unequal to 1, we show below that {{Mvar|f}} is invertible when considered as a function from the reals to the positive reals.

===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'').

=== Characterization by the product formula ===
The function {{math|log<sub>''b''</sub>&thinsp;''x''}} can also be essentially characterized by the product formula
<math display="block">\log_b(xy) = \log_b x + \log_b y.</math>
More precisely, the logarithm to any base {{math|''b'' > 1}} is the only [[increasing function]] ''f'' from the positive reals to the reals satisfying {{math|1=''f''(''b'') = 1}} and<ref>{{citation| title=Foundations of Modern Analysis |volume=1 |last=Dieudonné |first=Jean |page=84 |year=1969 |publisher=Academic Press }} item (4.3.1)</ref><math display="block">f(xy)=f(x)+f(y).</math>

===Graph of the logarithm function===
[[File:Logarithm inversefunctiontoexp.svg|right|thumb|The graph of the logarithm function {{math|log<sub>''b''</sub>&thinsp;(''x'')}} (blue) is obtained by [[Reflection (mathematics)|reflecting]] the graph of the function {{math|''b''<sup>''x''</sup>}} (red) at the diagonal line ({{math|1=''x'' = {{mvar|y}}}}).|alt=The graphs of two functions.]]
As discussed above, the function {{math|log<sub>''b''</sub>}} is the inverse to the exponential function <math>x\mapsto b^x</math>. Therefore, their [[graph of a function|graphs]] correspond to each other upon exchanging the {{mvar|x}}- and the {{mvar|y}}-coordinates (or upon reflection at the diagonal line {{Math|1=''x'' = ''y''}}), as shown at the right: a point {{math|1=(''t'', ''u'' = {{mvar|b}}<sup>''t''</sup>)}} on the graph of {{Mvar|f}} yields a point {{math|1=(''u'', ''t'' = log<sub>''b''</sub>&thinsp;''u'')}} on the graph of the logarithm and vice versa. As a consequence, {{math|log<sub>''b''</sub>&thinsp;(''x'')}} [[divergent sequence|diverges to infinity]] (gets bigger than any given number) if {{mvar|x}} grows to infinity, provided that {{mvar|b}} is greater than one. In that case, {{math|log<sub>''b''</sub>(''x'')}}  is an [[increasing function]]. For {{math|''b'' < 1}}, {{math|log<sub>''b''</sub>&thinsp;(''x'')}} tends to minus infinity instead. When {{mvar|x}} approaches zero, {{math|log<sub>''b''</sub>&thinsp;''x''}} goes to minus infinity for {{math|''b'' > 1}} (plus infinity for {{math|''b'' < 1}}, respectively).

===Derivative and antiderivative===
[[File:Logarithm derivative.svg|thumb|The graph of the [[natural logarithm]] (green) and its tangent at {{math|''x'' {{=}} 1.5}} (black)|alt=A graph of the logarithm function and a line touching it in one point.]]
Analytic properties of functions pass to their inverses.<ref name=LangIII.3 /> Thus, as {{math|1=''f''(''x'') = {{mvar|b}}<sup>''x''</sup>}} is a continuous and [[differentiable function]], so is {{math|log<sub>''b''</sub>&thinsp;''y''}}. Roughly, a continuous function is differentiable if its graph has no sharp "corners". Moreover, as the [[derivative]] of {{math|''f''(''x'')}} evaluates to {{math|ln(''b'') ''b''<sup>''x''</sup>}} by the properties of the [[exponential function]], the [[chain rule]] implies that the derivative of {{math|log<sub>''b''</sub>&thinsp;''x''}} is given by<ref name="LangIV.2"/><ref>{{citation |work=Wolfram Alpha |title=Calculation of ''d/dx(Log(b,x))'' |publisher=[[Wolfram Research]] |access-date=15 March 2011 |url=http://www.wolframalpha.com/input/?i=d/dx(Log(b,x)) }}</ref>
<math display="block">\frac{d}{dx} \log_b x = \frac{1}{x\ln b}. </math>
That is, the [[slope]] of the [[tangent]] touching the graph of the {{math|base-''b''}} logarithm at the point {{math|(''x'', log<sub>''b''</sub>&thinsp;(''x''))}} equals {{math|1/(''x'' ln(''b''))}}.

The derivative of {{Math|ln(''x'')}} is {{Math|1/''x''}}; this implies that {{Math|ln(''x'')}} is the unique [[antiderivative]] of {{math|1/''x''}} that has the value 0 for {{math|1=''x'' = 1}}. It is this very simple formula that motivated to qualify as "natural" the natural logarithm; this is also one of the main reasons of the importance of the [[E (mathematical constant)|constant&nbsp;{{Mvar|e}}]].

The derivative with a generalized functional argument {{math|''f''(''x'')}} is
<math display="block">\frac{d}{dx} \ln f(x) = \frac{f'(x)}{f(x)}.</math>
The quotient at the right hand side is called the [[logarithmic derivative]] of ''{{Mvar|f}}''. Computing {{math|''f<nowiki>'</nowiki>''(''x'')}} by means of the derivative of {{math|ln(''f''(''x''))}} is known as [[logarithmic differentiation]].<ref>{{Citation | last1=Kline | first1=Morris | author1-link=Morris Kline | title=Calculus: an intuitive and physical approach | publisher=[[Dover Publications]] | location=New York | series=Dover books on mathematics | isbn=978-0-486-40453-0 | year=1998}}, p.&nbsp;386</ref> The antiderivative of the [[natural logarithm]] {{math|ln(''x'')}} is:<ref>{{citation |work=Wolfram Alpha |title=Calculation of ''Integrate(ln(x))'' |publisher=Wolfram Research |access-date=15 March 2011 |url=http://www.wolframalpha.com/input/?i=Integrate(ln(x)) }}</ref>
<math display="block">\int \ln(x) \,dx = x \ln(x) - x + C.</math>
[[List of integrals of logarithmic functions|Related formulas]], such as antiderivatives of logarithms to other bases can be derived from this equation using the change of bases.<ref>{{Harvard citations|editor1-last=Abramowitz|editor2-last=Stegun|year=1972 |nb=yes|loc=p. 69}}</ref>

===Integral representation of the natural logarithm===
[[File:Natural logarithm integral.svg|right|thumb|The [[natural logarithm]] of ''{{Mvar|t}}'' is the shaded area underneath the graph of the function {{math|1=''f''(''x'') = 1/''x''}}.|alt=A hyperbola with part of the area underneath shaded in grey.]]
The [[natural logarithm]] of {{Mvar|t}} can be defined as the [[definite integral]]:

<math display="block">\ln t = \int_1^t \frac{1}{x} \, dx.</math>
This definition has the advantage that it does not rely on the exponential function or any trigonometric functions; the definition is in terms of an integral of a simple reciprocal. As an integral, {{math|ln(''t'')}} equals the area between the {{mvar|x}}-axis and the graph of the function {{math|1/''x''}}, ranging from {{math|1=''x'' = 1}} to {{math|1=''x'' = ''t''}}. This is a consequence of the [[fundamental theorem of calculus]] and the fact that the derivative of {{math|ln(''x'')}} is {{math|1/''x''}}. Product and power logarithm formulas can be derived from this definition.<ref>{{Citation|last1=Courant|first1=Richard|title=Differential and integral calculus. Vol. I|publisher=[[John Wiley & Sons]]|location=New York|series=Wiley Classics Library|isbn=978-0-471-60842-4|mr=1009558|year=1988}}, section III.6</ref> For example, the product formula {{math|1=ln(''tu'') = ln(''t'') + ln(''u'')}} is deduced as:

<math display="block">\begin{align}
\ln(tu) &= \int_1^{tu} \frac{1}{x} \, dx \\
& \stackrel {(1)} = \int_1^{t} \frac{1}{x} \, dx + \int_t^{tu} \frac{1}{x} \, dx \\
& \stackrel {(2)} = \ln(t) + \int_1^u \frac{1}{w} \, dw \\
&= \ln(t) + \ln(u).
\end{align}</math>

The equality&nbsp;(1) splits the integral into two parts, while the equality&nbsp;(2) is a change of variable ({{math|1=''w'' = {{mvar|x}}/''t''}}). In the illustration below, the splitting corresponds to dividing the area into the yellow and blue parts. Rescaling the left hand blue area vertically by the factor&nbsp;{{Mvar|t}} and shrinking it by the same factor horizontally does not change its size. Moving it appropriately, the area fits the graph of the function {{math|1=''f''(''x'') = 1/''x''}} again. Therefore, the left hand blue area, which is the integral of {{math|''f''(''x'')}} from {{Mvar|t}} to {{Mvar|tu}} is the same as the integral from 1 to {{Mvar|u}}. This justifies the equality&nbsp;(2) with a more geometric proof.

[[File:Natural logarithm product formula proven geometrically.svg|thumb|center|A visual proof of the product formula of the natural logarithm|alt=The hyperbola depicted twice. The area underneath is split into different parts.]]

The power formula {{math|1=ln(''t''<sup>''r''</sup>) = ''r'' ln(''t'')}} may be derived in a similar way:

<math display="block">\begin{align}
\ln(t^r) &= \int_1^{t^r} \frac{1}{x}dx \\
&= \int_1^t \frac{1}{w^r} \left(rw^{r - 1} \, dw\right) \\
&= r \int_1^t \frac{1}{w} \, dw \\
&= r \ln(t).
\end{align}</math>
The second equality uses a change of variables ([[integration by substitution]]), {{math|1=''w'' = {{mvar|x}}<sup>1/''r''</sup>}}.

The sum over the reciprocals of natural numbers,
<math display="block">1 + \frac 1 2 + \frac 1 3 + \cdots + \frac 1 n = \sum_{k=1}^n \frac{1}{k},</math>
is called the [[harmonic series (mathematics)|harmonic series]]. It is closely tied to the [[natural logarithm]]: as {{Mvar|n}} tends to [[infinity]], the difference,
<math display="block">\sum_{k=1}^n \frac{1}{k} - \ln(n),</math>
[[limit of a sequence|converges]] (i.e. gets arbitrarily close) to a number known as the [[Euler–Mascheroni constant]] {{math|1 = ''γ'' = 0.5772...}}. This relation aids in analyzing the performance of algorithms such as [[quicksort]].<ref>{{Citation|last1=Havil|first1=Julian|title=Gamma: Exploring Euler's Constant|publisher=[[Princeton University Press]]|isbn=978-0-691-09983-5|year=2003}}, sections 11.5 and 13.8</ref>

===Transcendence of the logarithm===
[[Real number]]s that are not [[Algebraic number|algebraic]] are called [[transcendental number|transcendental]];<ref>{{citation|title=Selected papers on number theory and algebraic geometry|volume=172|first1=Katsumi|last1=Nomizu|author-link=Katsumi Nomizu|location=Providence, RI|publisher=AMS Bookstore|year=1996|isbn=978-0-8218-0445-2|page=21|url={{google books |plainurl=y |id=uDDxdu0lrWAC|page=21}}}}</ref> for example, [[Pi|{{pi}}]] and ''[[e (mathematical constant)|e]]'' are such numbers, but <math>\sqrt{2-\sqrt 3}</math> is not. [[Almost all]] real numbers are transcendental. The logarithm is an example of a [[transcendental function]]. The [[Gelfond–Schneider theorem]] asserts that logarithms usually take transcendental, i.e. "difficult" values.<ref>{{Citation|last1=Baker|first1=Alan|author1-link=Alan Baker (mathematician)|title=Transcendental number theory|publisher=[[Cambridge University Press]]|isbn=978-0-521-20461-3|year=1975}}, p.&nbsp;10</ref>