Foundation · Explanatory preview
Logarithms and product-to-sum relations
A logarithm is the inverse of exponentiation, converting state-count multiplications into additive entropy, and carrying historical notation distinctions.
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Why do logarithms transform multiplicative probabilities into additive thermodynamic quantities?
The natural logarithm ln(x) is the inverse of the exponential function: ln(e^u) = u and e^(ln x) = x. Its defining algebraic property is that it converts products into sums: ln(A * B) = ln(A) + ln(B), and powers into products: ln(f^n) = n * ln(f).
In 1905 (§5 of the light-quanta paper), Einstein proved that if entropy S is an additive state function for independent systems (S = S_1 + S_2) while the statistical state probability W is multiplicative (W = W_1 * W_2), then the connection S = phi(W) must satisfy phi(W_1 * W_2) = phi(W_1) + phi(W_2). The only continuous solution is the logarithm: S = k * ln(W) + constant.
For n = 10 independent molecules each occupying a volume fraction f = 1/2, the joint probability is (1/2)^10 and its logarithm is 10 * ln(1/2) approx -6.931472. Inverting Wien's radiation law u = A nu^3 exp(-beta nu / T) to solve for inverse temperature requires taking the logarithm: ln(u / (A nu^3)) = -beta nu / T.
Notation note: In 1905 German scientific literature (including Annalen der Physik), 'lg' denoted the natural logarithm with base e. Modern ISO notation reserves 'lg' for the common base-10 logarithm log_10 and uses 'ln' for the natural logarithm. For example, 1905 printed 'lg 2' meant ln(2) approx 0.693147, not log_10(2) approx 0.301030.
One worked example
Entropy of combined independent systems equals Boltzmann constant times natural log of product of state weights, which equals sum of individual entropies.
Natural log of 2 is ln(2) approx 0.693147, whereas common log of 2 is log_10(2) approx 0.301030. For W_1 = 4 and W_2 = 8, W = 32: ln(4) approx 1.386294, ln(8) approx 2.079442, and ln(32) approx 3.465736 = 1.386294 + 2.079442.
A stopping point: The natural logarithm is the unique continuous function mapping independent product states to additive thermodynamic quantities; 1905 printed 'lg' denotes the natural logarithm.
Interactive construction: turning multiplication into addition
In section 5 of the light-quanta paper, Einstein reasons about the entropy S of independent systems. When two independent systems with microstate counts W₁ and W₂ are combined, the total number of configurations multiplies: W = W₁ · W₂. However, the thermodynamic entropy must add: S = S₁ + S₂. The only continuous function satisfying φ(W₁ · W₂) = φ(W₁) + φ(W₂) is the logarithm: S = k ln W + const.
Notation display toggle:
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).
| State W₁ | State W₂ | Product W₁ · W₂ | ln(W₁) | ln(W₂) | Sum ln(W₁) + ln(W₂) | ln(W₁ · W₂) |
|---|---|---|---|---|---|---|
| 2 | 4 | 8 | 0.693147 | 1.386294 | 2.079442 | 2.079442 |
| 3 | 5 | 15 | 1.098612 | 1.609438 | 2.708050 | 2.708050 |
| 10 | 10 | 100 | 2.302585 | 2.302585 | 4.605170 | 4.605170 |
Textual summary of the construction
The table demonstrates that for any pair of numbers, the logarithm of their product exactly equals the sum of their individual logarithms. This algebraic homomorphism bridges statistical mechanics (where independent configurations multiply) and macroscopic thermodynamics (where entropy is an extensive, additive quantity). When reading 1905 papers, readers must translate printed “lg” to natural “ln” to obtain the correct physical entropies.
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