In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number x to the base b is the exponent to which b must be raised, to produce x. For example, since 1000 = 103, the logarithm base 10 of 1000 is 3, or log10 (1000) = 3. The logarithm of x to base b is denoted as logb (x), or without parentheses, logb x, or even without the explicit base, log x, when no confusion is possible, or when the base does not matter such as in big O notation. The logarithm base 10 is called the decimal or common logarithm and is commonly used in science and engineering. The natural logarithm has the number e ≈ 2.718 as its base; its use is widespread in mathematics and physics, because of its very simple derivative. The binary logarithm...

The logarithm log_bx for a base b and a number x is defined to be the inverse function of taking b to the power x, i.e., b^x. Therefore, for any x and b, x=log_b(b^x), (1) or equivalently, x=b^(log_bx). (2) For any base, the logarithm function has a singularity at x=0. In the above plot, the blue curve is the logarithm to base 2 (log_2x=lgx), the black curve is the logarithm to base e (the natural logarithm log_ex=lnx), and the red curve is the logarithm to base 10 (the common...

inverse of the exponential function, which maps products to sums

An equal temperament is a musical temperament or tuning system that approximates just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, which gives an equal perceived step size, as pitch is perceived roughly as the logarithm of frequency.In classical music and Western music in general, the most common tuning system since the 18th century has been 12 equal temperament (also known as 12-tone equal temperament, 12-TET or 12-ET, informally abbreviated as 12 equal), which divides the octave into 12 parts, all of which are equal on a logarithmic scale, with a ratio equal to the 12th root of 2 (12√2 ≈ 1.05946). That resulting smallest interval, 1⁄12 the width of an octave, is called a semitone or half step. In Western countries the term equal temperament, without qualification, generally means 12-TET. In modern times, 12-TET is usually tuned relative to a standard pitch of 440 Hz, called A440...

We have analyzed archival HST F606W images of 18 dwarf elliptical (dE) galaxy candidates in the Coma Cluster. We model the full radial extent of their light-profiles by simultaneously fitting a PSF-convolved Sersic R^(1/n) model and, when necessary, either a central point-source or a central PSF-convolved Gaussian. The luminosities of the central component L_nuc scale with the host galaxy luminosity L_gal such that L_nuc = 10^(4.76 +/- 0.10)*(L_gal/10^7)^(0.87 +/- 0.26). The underlying host galaxies display systematic departures from an exponential model that are correlated with the model-independent host galaxy luminosity and are not due to biasing from the nuclear component. The Pearson correlation coefficient between log(n) and central galaxy surface brightness mu_0 (excluding the flux from extraneous central components) is -0.83 at a significance level of 99.99%. The Pearson correlation coefficient between the logarithm of the Sersic index `n' and the host galaxy magnitude is -0.77 at a significance of 99.9%. We explain the observed relationship between dE galaxy luminosity and inner logarithmic profile slope (gamma-prime) as a by-product of the correlation between luminosity and Sersic index n. Including, from the literature, an additional 232 dE and E galaxies spanning 10 mag in absolute magnitude (M), the dE galaxies are shown to display a continuous sequence with the brighter E galaxies such that mu_0 brightens linearly with M until core formation causes the most luminous (M_B < -20.5 mag) E galaxies to deviate from this relation. The different behavior of dE and E galaxies in the M-mu_e (and M-_e) diagram, and the _e-log(R_e) diagram have nothing to do with core formation, and are in fact expected from the continuous and linear relation between M and mu_0 and M and log(n).