View source for Logarithmic differentiation

{{Short description|Method of mathematical differentiation}}
{{For|the derivative|Logarithmic derivative}}
{{Calculus}}

In [[calculus]], '''logarithmic differentiation'''  or '''differentiation by taking logarithms''' is a method used to [[derivative|differentiate]] [[function (mathematics)|functions]] by employing the [[logarithmic derivative]] of a function {{math|''f''}},<ref>{{cite book| title=Calculus demystified| pages=170| first=Steven G.|last=Krantz | publisher=McGraw-Hill Professional| year=2003 | isbn=0-07-139308-0}}</ref>
<math display="block">(\ln f)' = \frac{f'}{f} \quad \implies \quad  f' = f \cdot (\ln f)'.</math>

The technique is often performed in cases where it is easier to differentiate the logarithm of a function rather than the function itself. This usually occurs in cases where the function of interest is composed of a product of a number of parts, so that a logarithmic transformation will turn it into a sum of separate parts (which is much easier to differentiate). It can also be useful when applied to functions raised to the power of variables or functions. Logarithmic differentiation relies on the [[chain rule]] as well as properties of [[logarithms]] (in particular, the [[natural logarithm]], or the logarithm to the base ''[[e (mathematics)|e]]'') to transform products into sums and divisions into subtractions.<ref>{{cite book| title=Golden Differential Calculus | pages=282|author=N.P. Bali| publisher=Firewall Media | year=2005 | isbn=81-7008-152-1}}</ref><ref name="Bird">{{cite book|title=Higher Engineering Mathematics | first=John|last=Bird|pages=324 | publisher=Newnes |year=2006 | isbn=0-7506-8152-7}}</ref> The principle can be implemented, at least in part, in the differentiation of almost all [[differentiable function]]s, providing that these functions are non-zero.

==Overview==

The method is used because the properties of logarithms provide avenues to quickly simplify complicated functions to be differentiated.<ref>{{cite book| title=Calculus, single variable | first=Brian E.|last=Blank | pages=457| publisher=Springer| year=2006| isbn=1-931914-59-1}}</ref> These properties can be manipulated after the taking of natural logarithms on both sides and before the preliminary differentiation. The most commonly used logarithm laws are<ref name="Bird" />
<math display="block">\ln(ab) = \ln(a) + \ln(b), \qquad
\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b), \qquad
\ln(a^n) = n\ln(a).</math>

===Higher order derivatives===
Using [[Faà di Bruno's formula]], the n-th order logarithmic derivative is,
<math display="block">\frac{d^n}{dx^n} \ln f(x)
= \sum_{m_1+2m_2+\cdots+nm_n=n} \frac{n!}{m_1!\,m_2!\,\cdots\,m_n!} \cdot
\frac{(-1)^{m_1+\cdots+m_n-1} (m_1 +\cdots + m_n-1)!}{f(x)^{m_1+\cdots+m_n}} \cdot
\prod_{j=1}^n \left(\frac{f^{(j)}(x)}{j!}\right)^{m_j}.</math>
Using this, the first four derivatives are,
<math display="block">\begin{align}
 \frac{d^2}{dx^2} \ln f(x) &= \frac{f''(x)}{f(x)} - \left(\frac{f'(x)}{f(x)} \right)^2 \\[1ex]
 \frac{d^3}{dx^3} \ln f(x) &= \frac{f^{(3)}(x)}{f(x)} - 3 \frac{f'(x) f''(x)}{f(x)^2} + 2 \left(\frac{f'(x)}{f(x)} \right)^3 \\[1ex]
 \frac{d^4}{dx^4} \ln f(x) &= \frac{f^{(4)}(x)}{f(x)} - 4 \frac{f'(x) f^{(3)}(x)}{f(x)^2} - 3 \left(\frac{f''(x)}{f(x)}\right)^2 + 12 \frac{f'(x)^2 f''(x)}{f(x)^3} - 6 \left(\frac{f'(x)}{f(x)} \right)^4
\end{align}</math>

==Applications==

===Products===
{{Main|Product rule}}
A [[natural logarithm]] is applied to a product of two functions
<math display="block">f(x) = g(x) h(x)</math>
to transform the product into a sum
<math display="block">\ln(f(x))=\ln(g(x)h(x)) = \ln(g(x)) + \ln(h(x)). </math>
Differentiating by applying the [[chain rule|chain]] and the [[sum rule in differentiation|sum]] rules yields
<math display="block">\frac{f'(x)}{f(x)} = \frac{g'(x)}{g(x)} + \frac{h'(x)}{h(x)},</math>
and, after rearranging, yields<ref>{{cite book | title=An Elementary Treatise on the Differential Calculus | first=Benjamin|last=Williamson | publisher=BiblioBazaar, LLC | year=2008 | pages=25–26 | isbn=978-0-559-47577-1}}</ref>
<math display="block">f'(x) = f(x)\times \left\{\frac{g'(x)}{g(x)} + \frac{h'(x)}{h(x)}\right\} =
g(x) h(x) \times \left\{\frac{g'(x)}{g(x)} + \frac{h'(x)}{h(x)}\right\} = g'(x) h(x) + g(x) h'(x),</math>
which is the [[product rule]] for derivatives.

===Quotients===
{{Main|Quotient rule}}
A [[natural logarithm]] is applied to a quotient of two functions
<math display="block">f(x) = \frac{g(x)}{h(x)}</math>
to transform the division into a subtraction
<math display="block">\ln(f(x)) = \ln\left(\frac{g(x)}{h(x)}\right) = \ln(g(x)) - \ln(h(x))</math>
Differentiating by applying the [[chain rule|chain]] and the [[sum rule in differentiation|sum]] rules yields
<math display="block">\frac{f'(x)}{f(x)} = \frac{g'(x)}{g(x)} - \frac{h'(x)}{h(x)},</math>
and, after rearranging, yields
<math display="block">f'(x) = f(x) \times \left\{\frac{g'(x)}{g(x)} - \frac{h'(x)}{h(x)}\right\} =
\frac{g(x)}{h(x)} \times \left\{\frac{g'(x)}{g(x)} - \frac{h'(x)}{h(x)}\right\} = \frac{g'(x) h(x) - g(x) h'(x)}{h(x)^2},</math>

which is the [[quotient rule]] for derivatives.

===Functional exponents===
For a function of the form
<math display="block">f(x) = g(x)^{h(x)}</math>
the [[natural logarithm]] transforms the exponentiation into a product
<math display="block">\ln(f(x)) = \ln\left(g(x)^{h(x)}\right) = h(x) \ln(g(x))</math>
Differentiating by applying the [[chain rule|chain]] and the [[sum rule in differentiation|product]] rules yields
<math display="block">\frac{f'(x)}{f(x)} = h'(x) \ln(g(x)) + h(x) \frac{g'(x)}{g(x)},</math>
and, after rearranging, yields
<math display="block">f'(x) = f(x)\times \left\{h'(x) \ln(g(x)) + h(x)\frac{g'(x)}{g(x)}\right\} =
g(x)^{h(x)} \times \left\{h'(x) \ln(g(x)) + h(x) \frac{g'(x)}{g(x)}\right\}.</math>
The same result can be obtained by rewriting ''f'' in terms of [[exponential function|exp]] and applying the chain rule.

====General case====
Using [[Multiplication#Capital pi notation|capital pi notation]], let
<math display="block">f(x) = \prod_i (f_i(x))^{\alpha_i(x)}</math>
be a finite product of functions with functional exponents.

The application of natural logarithms results in (with [[Summation#Capital sigma notation|capital sigma notation]])
<math display="block">\ln (f(x)) = \sum_i\alpha_i(x) \cdot \ln(f_i(x)),</math>
and after differentiation,
<math display="block">\frac{f'(x)}{f(x)} = \sum_i \left[\alpha_i'(x)\cdot \ln(f_i(x)) + \alpha_i(x) \cdot \frac{f_i'(x)}{f_i(x)}\right].</math>
Rearrange to get the derivative of the original function,
<math display="block">f'(x) = \overbrace{\prod_i (f_i(x))^{\alpha_i(x)}}^{f(x)} \times\overbrace{\sum_i\left\{\alpha_i'(x)\cdot \ln(f_i(x))+\alpha_i(x)\cdot \frac{f_i'(x)}{f_i(x)}\right\}}^{[\ln (f(x))]'}.</math>

==See also==

* {{annotated link|Darboux derivative}}
* {{annotated link|Generalizations of the derivative}}
* {{annotated link|Lie group}}
* {{annotated link|List of logarithm topics}}
* {{annotated link|List of logarithmic identities}}
* {{annotated link|Maurer–Cartan form}}

==Notes==
{{reflist}}

{{Calculus topics}}

[[Category:Differential calculus]]
[[Category:Logarithms]]