Difference of two numbers divided by the logarithm of their quotient
Three-dimensional plot showing the values of the logarithmic mean.
In mathematics , the logarithmic mean is a function of two non-negative numbers which is equal to their difference divided by the logarithm of their quotient .
This calculation is applicable in engineering problems involving heat and mass transfer .
Definition
The logarithmic mean is defined as:
M
lm
(
x
,
y
)
=
lim
(
ξ
,
η
)
→
(
x
,
y
)
η
−
ξ
ln
(
η
)
−
ln
(
ξ
)
=
{
x
if
x
=
y
,
y
−
x
ln
y
−
ln
x
otherwise,
{\displaystyle {\begin{aligned}M_{\text{lm}(x,y)&=\lim _{(\xi ,\eta )\to (x,y)}{\frac {\eta -\xi }{\ln(\eta )-\ln(\xi )}\\[6pt]&={\begin{cases}x&{\text{if }x=y,\\[2pt]{\dfrac {y-x}{\ln y-\ln x}&{\text{otherwise,}\end{cases}\end{aligned}
for the positive numbers x, y .
Inequalities
The logarithmic mean of two numbers is smaller than the arithmetic mean and the generalized mean with exponent one-third but larger than the geometric mean , unless the numbers are the same, in which case all three means are equal to the numbers.
x
y
≤
x
−
y
ln
x
−
ln
y
≤
x
+
y
2
≤
(
x
2
+
y
2
2
)
1
/
2
≤
(
x
3
+
y
3
2
)
1
/
3
for all
x
>
0
and
y
>
0.
{\displaystyle {\sqrt {xy}\leq {\frac {x-y}{\ln x-\ln y}\leq {\frac {x+y}{2}\leq \left({\frac {x^{2}+y^{2}{2}\right)^{1/2}\leq \left({\frac {x^{3}+y^{3}{2}\right)^{1/3}\qquad {\text{ for all }x>0{\text{ and }y>0.}
[1] [2] [3]
Toyesh Prakash Sharma generalizes the arithmetic logarithmic geometric mean inequality for any n belongs to the whole number as
x
y
(
ln
x
y
)
n
−
1
(
n
+
ln
x
y
)
≤
x
(
ln
x
)
n
−
y
(
ln
y
)
n
ln
x
−
ln
y
≤
x
(
ln
x
)
n
−
1
(
n
+
ln
x
)
+
y
(
ln
y
)
n
−
1
(
n
+
ln
y
)
2
{\displaystyle {\sqrt {xy}\ \left(\ln {\sqrt {xy}\right)^{n-1}\left(n+\ln {\sqrt {xy}\right)\leq {\frac {x(\ln x)^{n}-y(\ln y)^{n}{\ln x-\ln y}\leq {\frac {x(\ln x)^{n-1}(n+\ln x)+y(\ln y)^{n-1}(n+\ln y)}{2}
Now, for n = 0 :
x
y
(
ln
x
y
)
−
1
ln
x
y
≤
x
−
y
ln
x
−
ln
y
≤
x
(
ln
x
)
−
1
ln
x
+
y
(
ln
y
)
−
1
ln
y
2
x
y
≤
x
−
y
ln
x
−
ln
y
≤
x
+
y
2
{\displaystyle {\begin{array}{ccccc}{\sqrt {xy}\left(\ln {\sqrt {xy}\right)^{-1}\ln {\sqrt {xy}&\leq &{\dfrac {x-y}{\ln x-\ln y}&\leq &{\dfrac {x(\ln x)^{-1}\ln x+y(\ln y)^{-1}\ln y}{2}\\[4pt]{\sqrt {xy}&\leq &{\dfrac {x-y}{\ln x-\ln y}&\leq &{\dfrac {x+y}{2}\end{array}
This is the arithmetic logarithmic geometric mean inequality. similarly, one can also obtain results by putting different values of n as below
For n = 1 :
x
y
(
1
+
ln
x
y
)
≤
x
ln
x
−
y
ln
y
ln
x
−
ln
y
≤
x
(
1
+
ln
x
)
+
y
(
1
+
ln
y
)
2
{\displaystyle {\sqrt {xy}\left(1+\ln {\sqrt {xy}\right)\leq {\frac {x\ln x-y\ln y}{\ln x-\ln y}\leq {\frac {x(1+\ln x)+y(1+\ln y)}{2}
for the proof go through the bibliography.
Derivation
Mean value theorem of differential calculus
From the mean value theorem , there exists a value ξ in the interval between x and y where the derivative f ′ equals the slope of the secant line :
∃
ξ
∈
(
x
,
y
)
:
f
′
(
ξ
)
=
f
(
x
)
−
f
(
y
)
x
−
y
{\displaystyle \exists \xi \in (x,y):\ f'(\xi )={\frac {f(x)-f(y)}{x-y}
The logarithmic mean is obtained as the value of ξ by substituting ln for f and similarly for its corresponding derivative :
1
ξ
=
ln
x
−
ln
y
x
−
y
{\displaystyle {\frac {1}{\xi }={\frac {\ln x-\ln y}{x-y}
and solving for ξ :
ξ
=
x
−
y
ln
x
−
ln
y
{\displaystyle \xi ={\frac {x-y}{\ln x-\ln y}
Integration
The logarithmic mean can also be interpreted as the area under an exponential curve .
L
(
x
,
y
)
=
∫
0
1
x
1
−
t
y
t
d
t
=
∫
0
1
(
y
x
)
t
x
d
t
=
x
∫
0
1
(
y
x
)
t
d
t
=
x
ln
y
x
(
y
x
)
t
|
t
=
0
1
=
x
ln
y
x
(
y
x
−
1
)
=
y
−
x
ln
y
x
=
y
−
x
ln
y
−
ln
x
{\displaystyle {\begin{aligned}L(x,y)={}&\int _{0}^{1}x^{1-t}y^{t}\ \mathrm {d} t={}\int _{0}^{1}\left({\frac {y}{x}\right)^{t}x\ \mathrm {d} t={}x\int _{0}^{1}\left({\frac {y}{x}\right)^{t}\mathrm {d} t\\[3pt]={}&\left.{\frac {x}{\ln {\frac {y}{x}\left({\frac {y}{x}\right)^{t}\right|_{t=0}^{1}={}{\frac {x}{\ln {\frac {y}{x}\left({\frac {y}{x}-1\right)={}{\frac {y-x}{\ln {\frac {y}{x}\\[3pt]={}&{\frac {y-x}{\ln y-\ln x}\end{aligned}
The area interpretation allows the easy derivation of some basic properties of the logarithmic mean. Since the exponential function is monotonic , the integral over an interval of length 1 is bounded by x and y . The homogeneity of the integral operator is transferred to the mean operator, that is
L
(
c
x
,
c
y
)
=
c
L
(
x
,
y
)
{\displaystyle L(cx,cy)=cL(x,y)}
.
Two other useful integral representations are
1
L
(
x
,
y
)
=
∫
0
1
d
t
t
x
+
(
1
−
t
)
y
{\displaystyle {1 \over L(x,y)}=\int _{0}^{1}{\operatorname {d} \!t \over tx+(1-t)y}
and
1
L
(
x
,
y
)
=
∫
0
∞
d
t
(
t
+
x
)
(
t
+
y
)
.
{\displaystyle {1 \over L(x,y)}=\int _{0}^{\infty }{\operatorname {d} \!t \over (t+x)\,(t+y)}.}
Generalization
Mean value theorem of differential calculus
One can generalize the mean to n + 1 variables by considering the mean value theorem for divided differences for the n -th derivative of the logarithm.
We obtain
L
MV
(
x
0
,
…
,
x
n
)
=
(
−
1
)
n
+
1
n
ln
(
[
x
0
,
…
,
x
n
]
)
−
n
{\displaystyle L_{\text{MV}(x_{0},\,\dots ,\,x_{n})={\sqrt[{-n}]{(-1)^{n+1}n\ln \left(\left[x_{0},\,\dots ,\,x_{n}\right]\right)}
where
ln
(
[
x
0
,
…
,
x
n
]
)
{\displaystyle \ln \left(\left[x_{0},\,\dots ,\,x_{n}\right]\right)}
denotes a divided difference of the logarithm.
For n = 2 this leads to
L
MV
(
x
,
y
,
z
)
=
(
x
−
y
)
(
y
−
z
)
(
z
−
x
)
2
(
(
y
−
z
)
ln
x
+
(
z
−
x
)
ln
y
+
(
x
−
y
)
ln
z
)
.
{\displaystyle L_{\text{MV}(x,y,z)={\sqrt {\frac {(x-y)(y-z)(z-x)}{2{\bigl (}(y-z)\ln x+(z-x)\ln y+(x-y)\ln z{\bigr )}.}
Integral
The integral interpretation can also be generalized to more variables, but it leads to a different result. Given the simplex
S
{\textstyle S}
with
S
=
{
(
α
0
,
…
,
α
n
)
:
(
α
0
+
⋯
+
α
n
=
1
)
∧
(
α
0
≥
0
)
∧
⋯
∧
(
α
n
≥
0
)
}
{\displaystyle S=\{\left(\alpha _{0},\,\dots ,\,\alpha _{n}\right):\left(\alpha _{0}+\dots +\alpha _{n}=1\right)\land \left(\alpha _{0}\geq 0\right)\land \dots \land \left(\alpha _{n}\geq 0\right)\}
and an appropriate measure
d
α
{\textstyle \mathrm {d} \alpha }
which assigns the simplex a volume of 1, we obtain
L
I
(
x
0
,
…
,
x
n
)
=
∫
S
x
0
α
0
⋅
⋯
⋅
x
n
α
n
d
α
{\displaystyle L_{\text{I}\left(x_{0},\,\dots ,\,x_{n}\right)=\int _{S}x_{0}^{\alpha _{0}\cdot \,\cdots \,\cdot x_{n}^{\alpha _{n}\ \mathrm {d} \alpha }
This can be simplified using divided differences of the exponential function to
L
I
(
x
0
,
…
,
x
n
)
=
n
!
exp
[
ln
(
x
0
)
,
…
,
ln
(
x
n
)
]
{\displaystyle L_{\text{I}\left(x_{0},\,\dots ,\,x_{n}\right)=n!\exp \left[\ln \left(x_{0}\right),\,\dots ,\,\ln \left(x_{n}\right)\right]}
.
Example n = 2 :
L
I
(
x
,
y
,
z
)
=
−
2
x
(
ln
y
−
ln
z
)
+
y
(
ln
z
−
ln
x
)
+
z
(
ln
x
−
ln
y
)
(
ln
x
−
ln
y
)
(
ln
y
−
ln
z
)
(
ln
z
−
ln
x
)
.
{\displaystyle L_{\text{I}(x,y,z)=-2{\frac {x(\ln y-\ln z)+y(\ln z-\ln x)+z(\ln x-\ln y)}{(\ln x-\ln y)(\ln y-\ln z)(\ln z-\ln x)}.}
Connection to other means
Arithmetic mean :
L
(
x
2
,
y
2
)
L
(
x
,
y
)
=
x
+
y
2
{\displaystyle {\frac {L\left(x^{2},y^{2}\right)}{L(x,y)}={\frac {x+y}{2}
Geometric mean :
L
(
x
,
y
)
L
(
1
x
,
1
y
)
=
x
y
{\displaystyle {\sqrt {\frac {L\left(x,y\right)}{L\left({\frac {1}{x},{\frac {1}{y}\right)}={\sqrt {xy}
Harmonic mean :
L
(
1
x
,
1
y
)
L
(
1
x
2
,
1
y
2
)
=
2
1
x
+
1
y
{\displaystyle {\frac {L\left({\frac {1}{x},{\frac {1}{y}\right)}{L\left({\frac {1}{x^{2},{\frac {1}{y^{2}\right)}={\frac {2}{\frac {1}{x}+{\frac {1}{y}
See also
References
Citations
Bibliography
Oilfield Glossary: Term 'logarithmic mean'
Weisstein, Eric W. "Arithmetic-Logarithmic-Geometric-Mean Inequality" . MathWorld .
Stolarsky, Kenneth B.: Generalizations of the logarithmic mean , Mathematics Magazine, Vol. 48, No. 2, Mar., 1975, pp 87–92
Toyesh Prakash Sharma.: https://www.parabola.unsw.edu.au/files/articles/2020-2029/volume-58-2022/issue-2/vol58_no2_3.pdf "A generalisation of the Arithmetic-Logarithmic-Geometric Mean Inequality , Parabola Magazine, Vol. 58, No. 2, 2022, pp 1–5