A mathematical function that is the inverse of exponentiation, used here to linearize brightness-distance relations.A logarithm is a mathematical function that answers the question: to what exponent must a fixed base be raised to produce a given number? For example, the base-10 logarithm of 1000 is 3, because 10^3 = 1000. In physics and astronomy, logarithms are indispensable for compressing vast ranges of values—such as brightness, distance, or frequency—into manageable scales. They also convert multiplicative relationships into additive ones, simplifying analysis.
Logarithms were introduced by John Napier in 1614 as a computational aid, and later refined by Henry Briggs. They were quickly adopted by astronomers like Johannes Kepler for calculations.
The logarithm of a number x with base b is written as log_b(x). It satisfies the identity b^(log_b(x)) = x. Common bases are 10 (common logarithm), e ≈ 2.718 (natural logarithm), and 2 (binary logarithm). In astronomy, magnitudes are defined on a logarithmic scale: a difference of 5 magnitudes corresponds to a factor of 100 in brightness. This makes it easy to compare stars that differ by billions in luminosity.
Logarithms are used in the magnitude scale for stars, in the decibel scale for sound, in the pH scale for acidity, and in the Richter scale for earthquakes.
The word 'logarithm' comes from Greek meaning 'ratio-number', because Napier originally conceived them as a way to turn multiplication into addition.
| D_{KL} | variable | Kullback-Leibler divergence — information divergence |
| P(x) | variable | probability density of distribution P — probability density |
| Q(x) | variable | probability density of distribution Q — probability density |
| x | variable | random variable — random variable |
| \int | operator | sums the integrand over all values of x — integral operator |
| \log | operator | computes the logarithm of the ratio — logarithm |
| P(k) | variable | Perturbed power spectrum — power spectrum cubic metre |
| P_0(k) | variable | Background power spectrum — power spectrum cubic metre |
| A | variable | Amplitude of the bump — amplitude |
| k | variable | Wavenumber — wavenumber per metre |
| k_0 | variable | Central wavenumber of the bump — wavenumber per metre |
| σ | variable | Width of the bump in log-k — width |
| exp | operator | Exponential function — exponential |
| log | operator | Natural logarithm — logarithm |
| x | variable | data sample — sample |
| z | variable | latent variable sample — latent variable |
| D(x) | variable | discriminator output for real data — discriminator output |
| G(z) | variable | generator output for latent variable — generator output |
| V(D,G) | variable | value function of the game — value function |
| min_G | operator | minimize with respect to generator parameters — minimization over generator |
| max_D | operator | maximize with respect to discriminator parameters — maximization over discriminator |
| E_{x~p_data} | operator | expectation over the empirical data distribution — expectation over data distribution |
| E_{z~p_z} | operator | expectation over the latent prior distribution — expectation over latent distribution |
| log | operator | natural logarithm — logarithm |
| P(D|θ) | variable | probability of data given parameters — probability |
| y_model | variable | model prediction — value |
| y_data | variable | observed data point — value |
| σ | variable | standard deviation of Gaussian noise — standard deviation |
| log | operator | natural logarithm — logarithm |
| ∑ | operator | sum over all data points — summation |
| [X/H] | variable | Logarithmic abundance of element X relative to hydrogen — logarithmic abundance ratio |
| N_X | variable | Number density of element X — number density reciprocal cubic metre |
| N_H | variable | Number density of hydrogen — number density reciprocal cubic metre |
| log | operator | Base-10 logarithm — logarithm |
