Logarithmic functions

sewer Napier is the gentlemans gentleman credited to get to contributed hugely to the fields of science, philosophy and mathematics. Many desire that he is the brainchild of the modern computer science since he helped in making propagation, division and root extraction much easier especially for in truth large poesy. In the world of mathematics the genius of a man, legerdemain Napier is credited to have invented the logs as early as 1614 and states in his withstand The Descriptio that he started contemplating the idea of logarithms twenty forms earlier which was in the year 1594. Using Napiers get across in his book, calculations were made use the logarithm identities. These are the present first and second laws of logarithms lumber XY = log X + Log Y as well asLog X / Y =Log X Log Y. In his book the DescriptioJohn Napier delimitate logarithmic function as a differential equation.When the grip is b and the multivariate is x the logarithm to the paper b of the var iable x can be defined as the power to which you would maturate b to get x. Other scientists define logarithm as the exp one(a)nt to which the base must be raised to produce a given number(Standler, B.R 1990). That is verbalized as if Logbx = n the bn = x or if Y = bLogx = by = x. there are three laws of logarithm that scientists practice session in interpreting logarithm These laws areThe product to sum radiation pattern This law expresses that the product of a logarithm is equal to the sum of the psyche logarithms and is denotative as Log bXY = Log b X+ Log b Y The second law The quotient of different rule states that the logarithm of a quotient is the same as subtracting the logarithm of the denominator from the logarithm of the numerator Logbx/y = Log bx Logby The third and final law The power rule states that logarithm of x equals to the exponent of that power multiplied to the logarithm of x Log bXn =nLogb XCommon logarithmsAs earlier identified a logarithm to be val id must contain a base and a variable. Logarithms are classified into two Natural logarithms and Common logarithm. In common logarithms the base of the logarithm is assumed to be 10 when not steerd in a function, that is log 100 = 2 if the base is not indicated since if log 10100 = x therefore 10x = 100 hence x = 2. Common logarithm is much than prevalent when using arithmetic series as opposed to geometrical series.Natural logarithmsIn the common logarithm system the base is expressed as b whereas in natural logarithms the base number is expressed as e. This number e comes into use after the cracking mathematician from Switzerland by the name Leonhard Euler. Currently e is the base utilize in calculus and has since been named as natural base. The value e flush toilet be puzzle outd from a series of factorials starting from one (1)This is e = 1 + 1/1 + +1/3 + and from this, the value of e is approximately 2.71828182845904. Currently, when Mathematicians calculate the natura l logarithm of a number they indicate it as (log x) whereas physicists and engineers denote natural logarithms as lnX. Therefore log eX=ln X(Olds, C.D.1963)Logarithms make multiplication and division easier especially when using very better-looking number, very small numbers and those with decimal points. Scientists use of the 1st and second laws of logarithms when adding the logarithms of the numbers the result is the logarithm of the product of those numbers whereas. Subtracting the logarithms of two numbers gives the logarithm of the quotient of the numbers.These arithmetic properties of logarithms make such calculations much speedy and slight laborious. Although logarithm table are slowly becoming obsolete collectible to the invention of calculators and computers, logarithms themselves are still very useful. However, for manual calculations which also take on a great degree of precision the logarithm tables are easier since one only needs to look up in the logarithm table and do some summation which are faster and easier than performing multiplication (Weisstein, E.W 2007).Other than making calculations less(prenominal) labor intensive and much faster the use of logarithms also increases the accuracy of the results of calculations. This is because the use of logarithms allows minimal breaks as the value in the table are approximations of the actual values and thus the error is spread.The Keplers Rudolphine table that was published in 1627, made use of the logarithms and this resulted in more accurate values of latitudes of stars. They also together with Napiers Analogues made it cheaper and easier to calculate angles and sides of spherical triangles. The importance of this new technique is evidenced by the development of logarithmic methods based on logarithmic scales enables multiplication to be fast(a) and easy since there is decreased long multiplication.Logarithms are very indispensable in the work of astronomists, navigators, mathematicians and all other scientific fields want chemistry and physics.Logarithms for chemists Chemists use logarithms to calculate chemical reactions that are ever occurring in the world that we are living in. for instance the measure of acidity of a substance is made easier when using logarithms. In the PH scale substances have PH ranging from 0 7. A juice with PH of 4 is 10 times more acidic that the one with a PH of 5. This PH scale is logarithmic and when there is a PH change of 1 unit the acidity changes by factor of 10. As identified by students of chemistry the strength of the acidity changes towards the negative direction that is the higher the PH, the less acidic the solution.This was calculated by use of very small numbers such as 0.00001 that is written in logarithmic form as (1 x 10-5) where 5 is the logarithm of the number (Standler B.R.1990). As we all contend acidic solutions contain hydrogen ions H+(aq) and the pH is found by measuring the logarithm of the concentration of th ese ions and since many people would be deep in thought(p) by negative numbers, the PH is written assuming the negative attribute and this not withstanding, the PH is a logarithmic scale and the acidity of a solution with a given PH is different from that of the next pH number not by 1unit but by factor 10. electrical and Electronic engineers use decibels and bels as units of measurements. The bell is devised in a expedient way to measure power loss in a forebode system wiring rather than giving in amplifiers originally, the bel used to represent the amount of signal power loss due to subway system over a standard length of electrical cable, however, it is presently defined in terms of logarithms of base 10. The Richter scale that is used to measure the quake intensity is a perfect analogy of the bel scale. The 6.0 Richter earthquakes are 10 times more powerful than a 5.0 Richter earthquake. This means that an advantage of using a logarithmic measurement scale is the tremendo us range of appendix affordable by a relatively small span of many values.ReferenceStrandler, R.B 1990 Editorial Mathematics for engineers. The journal ofUndergraduate mathematics and its finish vol II, pages 1-6, springOlds, C, D, 1963. Continued fractions, Random House New YorkWeisstein, Eric W. Natural logarithm from math world a wolfram web resourceAccessed online on 23/09/07