Annus MirabilisEinstein in 1905
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Foundation lesson

Logarithms: turning products into sums

A logarithm turns multiplying into adding. When two systems are independent, the probabilities of their states multiply while their entropies add, so entropy has to be a logarithm of probability.

Written for this edition, not translated from Einstein.

Why does entropy need a logarithm?

Adding is easier than multiplying, and a logarithm turns one into the other. The natural logarithm of 2 is about 0.693, and of 4 about 1.386. Add them and you get 2.079, which is the natural logarithm of 8, the product of 2 and 4.

The natural logarithm ln x answers one question: to what power must e, about 2.718, be raised to give x? Raised to 0.693, e gives 2, and raised to 1.386 it gives 4. Multiplying two powers of e adds the powers, so raised to 2.079 it gives 8. Written for any numbers, the logarithm of a product is the sum of the logarithms, and a power becomes a multiple:

ln⁡(A B)=ln⁡(A)+ln⁡(B)\ln\left(A\,B\right) = \ln\left(A\right) + \ln\left(B\right)
ln⁡(fn)=n ln⁡(f)\ln\left(f^{n}\right) = n\,\ln\left(f\right)

The log of A times B is log A plus log B, and the log of f to the power n is n times log f.

Entropy needs exactly this. In §5 of the light-quanta paper, Einstein takes two independent systems. Their entropies add, S = S₁ + S₂, while the probabilities of their states multiply, W = W₁W₂. If entropy is some function φ of probability, then φ(W₁W₂) = φ(W₁) + φ(W₂), and he concludes that φ is a logarithm:

S−S0=RN ln⁡(W)S - S_{0} = \frac{R}{N}\,\ln\left(W\right)

S minus S nought equals R over N times the natural log of W.

R is the gas constant and N the number of molecules in a mole, so R/N is what is now called Boltzmann's constant. The step from φ(W₁W₂) = φ(W₁) + φ(W₂) to the logarithm needs φ to be a smooth function, which Einstein takes for granted.

Einstein prints lg for the natural logarithm, as German physics did in 1905. Today lg usually means the base-10 logarithm and ln the natural one. So a printed lg 2 means ln 2, about 0.693, not log₁₀ 2, about 0.301.

Worked exampleTen molecules found in half the volume

In §5 Einstein asks how probable it is that n molecules, each moving independently through a volume v₀, are all found at one moment in a part v of it. Each molecule is in that part with probability v/v₀, and independent probabilities multiply:

W=(vv0)nW = \left(\frac{v}{v_{0}}\right)^{n}

W equals v over v nought, to the power n.

  1. Take n = 10 molecules and v = v₀/2. Then W = (1/2)¹⁰ = 1/1024, about 0.000977.
  2. The logarithm turns the tenth power into ten times: ln W = 10 × ln(1/2) = 10 × (−0.693) = −6.93.
  3. So the entropy of the state with every molecule in one half is lower by 6.93 × R/N: 6.93 times Boltzmann's constant.

For a mole of gas, n = N, and the same formula gives S − S₀ = R ln(v/v₀). That is the entropy change thermodynamics gives for an ideal gas compressed from v₀ to v at constant temperature.

Try it: turning multiplication into addition

In §5 of the light-quanta paper, Einstein takes two independent systems. The probabilities of their states multiply, W = W₁ · W₂, while their entropies add, S = S₁ + S₂. A logarithm turns the one into the other, which is why he concludes that entropy is a constant times ln W, plus a constant. The table checks the rule on three pairs of numbers.

Notation display toggle:

Switches the table between the symbol printed in 1905, lg, and the modern ln.

1905 historical notation vs Modern ISO standard

In 1905 German scientific printing (including Annalen der Physik), the symbol lg denoted the natural logarithm (base e).

In modern ISO 80000-2 notation, ln denotes the natural logarithm, while lg is reserved for the common base-10 logarithm (log₁₀).

1905 printed “lg 2”: 0.693147, the natural logarithm ln 2.

Modern ISO “lg 2” (log₁₀ 2): 0.301030, the common base-10 logarithm.

ln(W₁ · W₂) = ln(W₁) + ln(W₂), checked on three pairs
W₁ and W₂ln(W₁) + ln(W₂)ln(W₁ · W₂)
2 and 4
product 8		0.693147
+ 1.386294
= 2.079442		ln 8
= 2.079442
3 and 5
product 15		1.098612
+ 1.609438
= 2.708050		ln 15
= 2.708050
10 and 10
product 100		2.302585
+ 2.302585
= 4.605170		ln 100
= 4.605170

What it shows, in words

For each of three pairs of numbers, the logarithm of the product equals the sum of the two logarithms. That is the property entropy needs: the probabilities of independent systems multiply and their entropies add. In the 1905 papers the logarithm is printed lg and means the natural logarithm, ln.

Where this lesson stops

A logarithm turns products into sums, which is why it connects probabilities that multiply to entropies that add. In the 1905 papers, lg means the natural logarithm.

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