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IT AND TECHNOLOGY: On the Discrete Logarithm Problem

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Logarithms have a correspondence story based on their history between positive real-number multiplication and addition on the real number line, which was formalized in the seventeenth century beginning in 1620. Logarithm tables have been produced in a variety of formats over the years. The slide rule was built using the concept of logarithms. The expression of area coming from a breakthrough in natural logarithm necessitated the incorporation of new functions resulting in mainstream mathematics (Stepanov). Thus, the study focuses on logarithms in music, computers, and their applications in the present world.

Logarithms help with converting ratios to cents or semitones.The language of mathematical set theory is applied by musical set theory in an elementary way to organize the various musical objects and also describe their relationships. The motives or cords formed help analyze the structure of a piece of typically atonal music. Isometrics; transposition and inversion, help discover deep structures of music (Aneke and Wang). Abstract algebra can be used to analyze music; tempered octave containing equal pitch classes form an Abelian group containing 12 elements. Just intonation can be described in terms of free Abelian group. The free and transitive action of the cyclic group Z/12Z contained by the chromatic scale, defines via transportation of notes. Consequently, for the group Z/12Z the chromatic scale can be thought of as torsos. A rhythmic structure, a fundamental of equal and regular arrangements of the repetition of the pulse, accent, phrase and duration, builds up the music (Zhang and Wang). A diatonic scale is one of the music scales and is made up of discrete set of pitches (Stepanov).

Logarithms are effectively used in the current modern world on the phrases based on six figures, double digits, order of magnitude and evaluation of the interest rates. Numbers are usually described in terms of their powers 10 which is the logarithm. In some arcane formula, Mathematics usually point out logarithms. Concepts in Mathematics are expressed with notation like In or log. The encounter of different ideas in life and seeing how they could be written with notation is basically applying mathematics. Logarithms usually find the cause for a particular effect based on the input of something for some output. For instance, if something grew from 50 to 100, then the continuous return of in is 8.1 % (Aneke and Wang). Logarithms put numbers on human friendly scale by writing the numbers in form of In. Based on the order of magnitude, the bits counted by a computer results to each bit having a double effect. For instance, 2 raised to power 48 will result to 256. Logarithms also assist in the determination of the interest rates. The GDP of the country’s one year and the next are taken, henceforth logarithm is applied in determining the implicit growth rate (Aneke and Wang).

Business, engineering, and science applications apply the logarithmic functions. Programing languages such as C use the logarithmic formulas in establishment of non-linear relationships and to define different number of inputs to calculations. For instance, the use of log 10 natural logarithms applied in numerical inputs results to functioning to the tenth power under the big development of Microsoft Windows products. Calculation of real arguments outputs applied by the logarithmic functions in a computer program must always be greater than zero (Cima et al.).

Logarithms are also related with computers based on their technology versions as the logarithmic exponents used in computer programmers simplify complex mathematical calculations. For instance, 10^4= 10000 and can also be described as 4 = log10000. Logarithms are applied on the computer function formulas to develop computer developers which may be used in comparison of different statistical data in the creation of graphs. The prediction of outcomes based on upon mathematical statistical information are created by computer modelling which develops the comparison. Logarithms may represent subjects for comparison such as the intensities triggered by earthquakes and brightness of light (Cima et al.). Different magnitudes measured by computers, builds up logarithmic scales. There is also an analysis of exponential processes such as the different spreads of various epidemics by the plotting sets of logarithmic measurements. A logistic is thus modified.

In addition to that, the alignment of pixels, organization of colors and manipulation of photographs for enhancement or comparison are applied by logarithms in the computers. The digital image created, the information photographed converts into smaller sections of color; pixels. The organization of the red (R), green (G), blue (B) values each pixel to transformation of color pairs such as G/R results to formation of an image. The specific mathematical logarithm are signified by each of the pairs allowing the translation and alignment of each pixel into the image photographed (Cima et al.).

In addition to that, effective computer cryptosystems are formed by Discrete Logarithms. Certain logarithmic formulas are exchanged by the variable nature of the numerical key. Thus allows cryptologists to form security systems based on the computers which restrict the user access. They also act as a sieve barring specific security attacks (Cima et al.).

In conclusion, logarithms focus on the inverse operations to various exponentiations. Logarithm has a wide application on music, computers and the current modern world. In music, logarithm enhances conversion of ratios into cents or semitones. The technology developed in computers applies logarithm in the display of images and development of security. Lastly, the current modern world, logarithm focuses on the phrases based on six figures, double digits, order of magnitude and evaluates the interest rates (Aneke and Wang).

Work Cited

Aneke, Mathew, and Meihong Wang. “Energy Storage Technologies and Real Life Applications – A State of the Art Review.” Applied Energy, vol. 179, 2016, pp. 350–77, doi:10.1016/j.apenergy.2016.06.097.

Cima, Anna, et al. “Computers and Mathematics with Applications.” Computers and Mathematics with Applications, vol. 60, no. 5, 2010, pp. 1457–64, doi:10.1016/j.camwa.2008.02.039.

Stepanov, Sergei A. “On the Discrete Logarithm Problem.” Discrete Mathematics and Applications, vol. 24, no. 1, 2014, pp. 45–52, doi:10.1515/dma-2014-0005.

Zhang, Fangguo, and Ping Wang. “Speeding up Elliptic Curve Discrete Logarithm Computations with Point Halving.” Designs, Codes, and Cryptography, vol. 67, no. 2, 2013, pp. 197–208, doi:10.1007/s10623-011-9599-5.

May 24, 2023
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