Logarithmic mean

Key takeaways

Logarithmic mean belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Logarithmic mean to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Logarithmic mean from memory before moving on to harder problems.

Reference excerpt

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 by

L ( x , y ) = { x , if x = y , x − y ln ⁡ x − ln ⁡ y , otherwise , {\displaystyle L(x,y)=\left\{{\begin{array}{l l}x,&{\text{if }}x=y,\\{\dfrac {x-y}{\ln x-\ln y}},&{\text{otherwise}},\end{array}}\right.}

for x , y ∈ R {\displaystyle x,y\in \mathbb {R} } , such that x , y > 0 {\displaystyle x,y>0} .

Inequalities The logarithmic mean of two numbers is smaller than the arithmetic mean and the generalized mean with exponent greater than 1. However, it is larger than the geometric mean and the harmonic mean, respectively. The inequalities are strict unless both numbers are equal. More precisely, for p , x , y ∈ R {\displaystyle p,x,y\in \mathbb {R} } with x ≠ y {\displaystyle x\neq y} and p > 1 {\displaystyle p>1} , we have

2 x y x + y < x y < x − y ln ⁡ x − ln ⁡ y < x + y 2 < ( x p + y p 2 ) 1 / p , {\displaystyle {\frac {2xy}{x+y}}<{\sqrt {xy}}<{\frac {x-y}{\ln x-\ln y}}<{\frac {x+y}{2}}<\left({\frac {x^{p}+y^{p}}{2}}\right)^{1/p},}

where the expressions in the chain of inequalities are, in order: the harmonic mean, the geometric mean, the logarithmic mean, the arithmetic mean, and the generalized arithmetic mean with exponent p {\displaystyle p} .

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 is also given by the integral

L ( x , y ) = ∫ 0 1 x 1 − t y t d t . {\displaystyle L(x,y)=\int _{0}^{1}x^{1-t}y^{t}\,\mathrm {d} t.}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Logarithmic mean

Start with the simplest possible case. Write down what Logarithmic mean claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Logarithmic mean before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Logarithmic mean ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Logarithmic mean

In research: Logarithmic mean appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Logarithmic mean in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Logarithmic mean is common in secondary-school and first-year university syllabi. It links to neighbouring topics Logarithms, Means, so understanding it makes those chapters shorter.
In everyday life: Look for Logarithmic mean outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Logarithmic mean in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Logarithmic mean means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Logarithmic mean out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Logarithmic mean in simple terms?

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.

Why does Logarithmic mean matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Logarithmic mean?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Logarithmic mean.