In mathematics , in particular calculus and complex analysis , the Logarithmic Derivative of a function f is defined by the formula
{\displaystyle {\frac {f'}{f}}}where is the derivative of f . Intuitively, this is a subtle relative change in f ; That is, the subtle absolute change in f , i.e. f’, Incremented by the current value of f .
When f is a function f ( x ) of a real variable x , and takes real , strictly positive values, this is equal to the derivative of ln ( f ), or the natural logarithm of f . It follows the straight chain rule .
{\displaystyle {\frac {d}{dx}}\ln f(x)={\frac {1}{f(x)}}{\frac {df(x)}{dx}}}
Core properties
Many properties of real logarithms also apply to logarithmic derivatives, even when the function does not take a value in a positive real. For example, since the logarithm of a product is the sum of the logarithms of the factors, we have
(\log uv)'=(\log u+\log v)'=(\log u)'+(\log v)'.\!
So for positive-real-valued functions, the logarithmic derivative of a product is the sum of the logarithmic derivatives of the factors. But we can also use Leibniz law for the derivative of a product
{\frac {(uv)'}{uv}}={\frac {u'v+uv'}{uv}}={\frac {u'}{u}}+{\frac {v'}{v}}.\!
Thus, it is true for any function that the log derivative of a product is the sum of the logarithmic derivatives of the factors (when they are defined).
One consequence of this is that the log derivative of the inverse of a function is the negation of the log derivative of the function:
{\frac {(1 / u) '} {1 / u}} = {\frac {-u' /u ^ {2}} {1 /u}} = - {\frac {u '} {u} }, \!
Just as the logarithm of the inverse of a positive real number is the negation of the logarithm of the number.
More generally, the logarithmic derivative of a quotient is the difference between the logarithmic derivative of the dividend and the divisor:
{\frac {(u/v)'}{u/v}}={\frac {(u'v-uv')/v^{2}}{u/v}}={\frac {u'}{u}}-{\frac {v'}{v}},\!
Just as the logarithm of a quotient is the difference between the logarithm of the divisor and the divisor.
Generalizing in another direction, the logarithmic derivative of a power (with a constant real exponent) is the product of the logarithmic derivative of the exponent and the base:
{\frac {(u ^ {k}) '} {u ^ {k}}} = {\frac {ku ^ {k-1} u'} {u ^ {k}}} = k {\frac { u '} {u}}, \!
Just as the logarithm of a power is the product of the exponent and the logarithm of the base.
In short, both derivatives and logarithms have a product rule , a reciprocal rule , a quotient rule , and a power rule ( compare List of logarithmic identities ); Each pair of rules is related by means of a logarithmic derivative.
Calculating Simple Derivatives Using Logarithmic Derivatives
Logarithmic derivatives can simplify the computation of derivatives requiring the product rule while producing similar results . The procedure is as follows: Let’s say ( x ) = u ( x ) v ( x ) and we want to calculate ‘( x ) . Instead of computing it directly as =’ = u’ v + v’ u , we calculate its logarithmic derivative. That is, we calculate:
{\frac {f'}{f}}={\frac {u'}{u}}+{\frac {v'}{v}}.
Multiplying by ‘ is calculate as :
{\displaystyle f'=f\cdot \left({\frac {u'}{u}}+{\frac {v'}{v}}\right).}This technique is most useful when when is the product of a large number of factors. This technique makes it possible to calculate the logarithmic derivative of each factor, sum and multiply by to calculate ‘ .
Integrating factors
The idea of logarithmic derivatives is closely linked to the integrating factor method for first-order differential equations . In operator terms, write
{\displaystyle D={\frac {d}{dx}}}And let M denote the operator of the multiplication by a given function G ( x ). Then
{\displaystyle M^{-1}DM}( by product rule ) can be written as:
{\displaystyle D+M^{*}}
where is now denotes the multiplication operator by the logarithmic derivative M*
{\displaystyle {\frac {G'}{G}}}
In practice we are given an operator like
{\displaystyle D+F=L}
and want to solve the equations
{\displaystyle L(h)=f}For the function h , f is given . It then reduces to solving
{\displaystyle {\frac {G'}{G}}=F}whose solution is
{\displaystyle \exp \textstyle (\int F)}With any indefinite integral of f .
Complex analysis
The given formula can be applied more broadly; For example, if f ( z ) is a meromorphic function , it makes sense for all complex values of z at which f has neither a zero nor a pole . Furthermore, the logarithmic derivative at a zero or a pole behaves in a way that is easily analyzed in terms of the special case. zn
With n an integer, n 0. The logarithmic derivative is thenn / z ;
And one can draw the general conclusion that for f meromorphic, the specificity of the logarithmic derivative of f are all simple poles, with residues n of order minus n , residues – n from a pole of order n . See logic theory . This information is often used in contour integration .
In the field of Nevanlinna theory , an important lemma states that, for example, the adjacency function of the logarithmic derivative with respect to the Nevanlinna characteristic of the original function is trivial .
m(r,h'/h)=S(r,h)=o(T(r,h))
Multiplier group
Behind the use of the logarithmic derivative are two basic facts about GL1 , which is the multiplicative set of real numbers or other fields . difference operator
X{\frac {d}{dX}}
is invariant under ‘translation’ (replacing x by a constant for ax ). and differential form dx/x
Likewise it is immutable. For the function F to GL 1 , the formula df/f
Therefore there is a pullback of the irreversible form .
Example
- Exponential growth and exponential decay are processes with a constant logarithmic derivative.
- In mathematical finance , the Greek derivative with respect to the underlying price is the logarithmic derivative of the price.
- In numerical analysis , the condition number is a small relative change in output to a relative change in input, and thus is the ratio of logarithmic derivatives.
