In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x.

374 Results for the subject "Logarithms":

Publisher SummaryThis chapter presents the applications of a new notation for exponents. The properties of logarithms are actually the properties of exponents. Logarithms were earlier used extensively to simplify tedious calculations. As handheld calculators are so common now, logarithms are seldom used in connection with computations. Nevertheless, there are many other applications of logarithms to both science and higher mathematics. A common logarithm is a logarithm with a base of 10. As common logarithms are used so frequently, it is customary, in order to save time, to omit notating the base. The chapter presents two special identities that arise from the definition of logarithms. Each is useful in proving the properties for evaluating some simple logarithmic expressions.

Charles P. McKeague Publication date: 1983/01/01AbstractThis chapter refreshes your knowledge of permutations and combinations as well as logarithms.

Julien I.E. Hoffman Publication date: 2019/01/01AbstractThe discrete logarithm over finite fields of small characteristic can be solved much more efficiently than previously thought. This algorithmic breakthrough is based on pinpointing relations among the factor base discrete logarithms. In this paper, we concentrate on the Kummer extension Fq2(q−1)=Fq2[x]/(xq−1−A). It has been suggested that in this case, a small number of degenerate relations (from the Borel subgroup) are enough to solve the factor base discrete logarithms. We disprove the conjecture, and design a new heuristic algorithm with an improved bit complexity O˜(q1+θ) (or algebraic complexity O˜(qθ)) to compute discrete logarithms of all the elements in the factor base {x+α|α∈Fq2}, where θ<2.38 is the matrix multiplication exponent over rings. Given additional time O˜(q4), we can compute discrete logarithms of at least Ω(q3) many monic irreducible quadratic polynomials. We reduce the correctness of the algorithm to a conjecture concerning the det *Read more...*

AbstractLaih and Kuo proposed two efficient signature schemes based on discrete logarithms and factorization. However, their schemes require many keys for a signing document. In this article, we shall propose an improvement of Laih and Kuo's signature schemes. The improved scheme will outperform their schemes in the number of keys.

Li-Hua Li Publication date: 2005/02/04AbstractLaih and Kuo proposed two efficient signature schemes based on discrete logarithms and factorization. Recently, Li et al. improve one of their schemes in order to use fewer keys for a signing document. In this paper, we shall prove that their improvement of Laih and Kuo’s signature scheme is insecure. Moreover the improved signature scheme in fact is not based on two cryptographic assumptions simultaneously, and forging a signature on any message would not need to solve any difficult problems.

Haifeng Qian Publication date: 2005/07/26AbstractThe matrix elements of operators containing both heavy quark (Q) and light quark (q) fields can contain large logarithms of the type ln(mQ2/μ2), where μ is a typical QCD mass scale and mQ is the heavy quark mass. We outline a method for summing leading logarithms of this type. We apply it to the decay constant fM of a low lying pseudoscalar meson M with Q̄q flavor quantum numbers and predict the ratios of decay constants for mesons with different heavy flavors. We also apply it to a matrix element of a four-quark operator which is relevant for B0−B̄0 mixing.

H.David Politzer Publication date: 1988/06/02AbstractWe show how perturbation theory may be reorganized to give splitting functions which include order by order convergent sums of all leading logarithms of x. This gives a leading twist evolution equation for parton distributions which sums all leading logarithms of x and Q2, allowing stable perturbative evolution down to arbitrarily small values of x. Perturbative evolution then generates the double scaling rise of F2 observed at HERA, while in the formal limit x → 0 at fixed Q2 the Lipatov xf-λ behaviour is eventually reproduced. We are thus able to explain why leading order perturbation theory works so well in the HERA region.

Richard D. Ball Publication date: 1995/05/25AbstractThe dominant coupling constant logarithms which arise in perturbatively resummed massless (QED)3, are functionally evaluated exactly and summed to all orders in various amplitudes. In addition to the perturbative sums, unexpected non-perturbative, non-Borel summable terms are encountered.

Stephen Templeton Publication date: 1981/07/16Publisher SummaryThis chapter reviews a differential-algebraic study of the intrusion of logarithms into asymptotic expansions. The nonappearances of high-ranking logarithms are elementary instances of results which are obtained in the chapter., The domain of operations is a graduated logarithmic field, which, roughly speaking, is an abstract differential field having extra structure designed to provide a setting for asymptotic expansions. The rank-rise problem for asymptotic expansions is to determine, for an r-inscribed first-order differential polynomial P, how large the logarithmic rank of mn can be if mn is a term in an asymptotic expansion of a solution of P.

Walter Strodt Publication date: 1977/01/01AbstractBy solving a differential-functional equation inposed by the MaxEnt principle we obtain a class of two-parameter deformed logarithms and construct the corresponding two-parameter generalized trace-form entropies. Generalized distributions follow from these generalized entropies in the same fashion as the Gaussian distribution follows from the Shannon entropy, which is a special limiting case of the family. We determine the region of parameters where the deformed logarithm conserves the most important properties of the logarithm, and show that important existing generalizations of the entropy are included as special cases in this two-parameter class.

G. Kaniadakis Publication date: 2004/09/01AbstractExponentials and logarithms are important basic mathematical functions. This chapter introduces the basic exponential and logarithmic functions and their properties.

Xin-She Yang Publication date: 2017/01/01AbstractNew lower bounds for linear forms in n (≥ 2) elliptic logarithms in the CM case are established. The estimate is better than all previous estimates with respect to some of the parameters that appear. It may be interesting to notice that the product log A1 … log An in the lower bound (see the Corollary of Theorem 1) is of exactly the same form as in the lower bounds for linear forms in logarithms of algebraic numbers (see A. Baker [in “Transcendence Theory: Advances and Applications”. (A. Baker and D. W. Masser, Eds.), pp. 1–27, Academic Press, New York, 1977]) and this is the first time such a parallelism has been achieved. To obtain the above lower bounds a zero estimate on the group variety Gan × E (Cn × E) is established (with E being an elliptic curve with CM), which is sharper than that derived from the general results in D. W. Masser and G. Wüstholz (Inventiones Math. 63 (1981), 81–95).

Kunrui Yu Publication date: 1985/02/01AbstractIn this paper, we develop parallel algorithms for integer factoring and for computing discrete logarithms. In particular, we give polylog depth probabilistic boolean circuits of subexponential size for both of these problems, thereby solving an open problem of Adleman and Kompella. Existing sequential algorithms for integer factoring and discrete logarithms use a prime base which is the set of all primes up to a bound B. We use a much smaller value for B for our parallel algorithms than is typical for sequential algorithms. In particular, for inputs of length n, by setting B = nlogdn with d a positive constant, we construct •Probabilistic boolean circuits of depth (log) and size exp[(/log)] for completely factoring a positive integer with probability 1−(1), and•Probabilistic boolean circuits of depth (log + log) and size exp[(/log)] for computing discrete logarithms in the finite field () for a prime with probability 1−(1).
These are the first results of this type for b *Read more...*

AbstractWe present a unified derivation of the resummation of Sudakov logarithms, directly from the factorization properties of cross sections in which they occur. We rederive in this manner the well-known exponentiation of leading and non-leading logarithmic enhancements near the edge of phase space for cross sections such as deeply inelastic scattering, which are induced by an electroweak hard scattering. The relevant factorization theorems are known to hold for many such cross sections of interest, and we conjecture that they apply even more widely. For QCD hard-scattering processes, such as heavy-quark production, we show that the resummation of non-leading logarithms requires in general mixing in the space of the color tensors of the hard scattering. The exponentiation of Sudakov logarithms implies that many weighted cross sections obey particular evolution equations in momentum transfer, which streamline the computation of their Sudakov exponents. We illustrate this method with t *Read more...*

AbstractWe investigate two specific Green functions in the framework of chiral perturbation theory. We show that, using analyticity and unitarity, their leading logarithmic singularities can be evaluated in the chiral limit to any desired order in the chiral expansion, with a modest calculational cost. The claim is illustrated with an evaluation of the leading logarithm for the scalar two-point function to five-loop order.

M. Bissegger Publication date: 2007/03/08AbstractWe examine the leading double logarithm structure in the calculation of jet fractions using the JADE algorithm, based on a jet-jet mass cut ys. We find that there is no simple formula allowing us to explicitly sum these logarithms, a necessary procedure if we are to apply perturbation theory in the region y⪡1 where these double logarithms are large. This casts doubt on the usefulness of such an algorithm for comparing the predictions of perturbative QCD with the experimental measurements at small y.

N. Brown Publication date: 1990/12/27AbstractUsing elementary Banach algebra techniques, it is determined which elements of Banach algebras like l1(α), b(K), and A have logarithms. The Wiener-Lévy theorem will be used to answer the same question for more complicated Banach algebras like the Wiener algebra. These results will be applied to polynomials of convolution type and a generalization thereof.

A. Di Bucchianico Publication date: 1991/03/15Publisher SummaryThis chapter focuses on the exponential and logarithmic functions. Exponential functions are useful in chemistry, biology, economics, mathematics, and engineering. The chapter describes the applications of exponential functions in calculating such quantities as compound interest and the growth rate of bacteria in a culture medium. Exponential functions occur in a wide variety of applied problems. The chapter reviews some problems dealing with population growth: predicting the growth of bacteria in a culture medium; radioactive decay, such as determining the half-life of strontium 90; and the interest earned when an interest rate is compounded. Logarithms can be viewed as another way of writing exponents. Logarithms have been used to simplify calculations; the slide rule, a device long used by engineers, is based on logarithmic scales. In today's world of inexpensive hand calculators, the need for manipulating logarithms is reduced.

Bernard Kolman Publication date: 1986/01/01AbstractWe present our predictions for the inclusive production of two heavy quark–antiquark pairs, separated by a large rapidity interval, in the collision of (quasi-)real photons at the energies of LEP2 and of some future electron–positron colliders. We include in our calculation the full resummation of leading logarithms in the center-of-mass energy and a partial resummation of the next-to-leading logarithms, within the Balitsky–Fadin–Kuraev–Lipatov (BFKL) approach.

F.G. Celiberto Publication date: 2018/02/10AbstractWe examine the emergence of the infrared logarithms in the cosmological perturbation theory applied to the minisuperspace scalar slow-roll inflation. Not surprisingly, in the single scalar field model the curvature perturbation ζ is conserved and no lnaB behavior appears, where aB(t) is the background scale factor of the universe. On the other hand, in the presence of a spectator scalar the n'th order perturbation theory gives an (lnaB)n correction to ζ. However, a nonperturbative estimate shows that ζ actually becomes the sum of a constant and a mildly evolving lnaB pieces.

Ali Kaya Publication date: 2018/07/10