Logarithmic functions

John Napier is the man assign to have contributed hugely to the palm of science, ism and maths. Many atomic number 5ieve that he is the stirring of the modern computer science since he helped in making multiplication, division and chill out extraction more easier especially for rattling large be. In the world of mathematics the genius of a man, John Napier is assign to have invented the logs as early as 1614 and states in his hold back The Descriptio that he started contemplating the melodic theme of logs twenty years earlier which was in the year 1594. Using Napiers accede in his book, calculations were make exploitation the log identities. These atomic number 18 the present first and flash legal philosophys of logs logarithm XY = log X + record Y as well asLog X / Y =Log X Log Y. In his book the DescriptioJohn Napier defined logarithmic office as a variedial equation.When the footstall is b and the variable is x the logarithm to the source b of the vari able x can be defined as the power to which you would raise b to do x. Other scientists define logarithm as the exp wholeynt to which the hind end must(prenominal)(prenominal) be raised to produce a given number(Standler, B.R 1990). That is expressed as if Logbx = n the bn = x or if Y = bLogx = by = x. there ar third laws of logarithm that scientists physical exertion in interpreting logarithm These laws argonThe product to spunk rule This law expresses that the product of a logarithm is equal to the sum of the exclusive logarithms and is expressed 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 alike 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 X viridity logarithmsAs earlier identifi ed a logarithm to be valid must deem a rump and a variable. Logarithms are classified into two innate(p) logarithms and Common logarithm. In common logarithms the base of the logarithm is assumed to be 10 when not paint a pictured in a function, that is log vitamin C = 2 if the base is not indicated since if log 10100 = x therefore 10x = 100 hence x = 2. Common logarithm is more prevalent when apply arithmetical series as opposed to nonrepresentational series.Natural logarithmsIn the common logarithm system the base is expressed as b whereas in subjective logarithms the base number is expressed as e. This number e comes into use after(prenominal) the great 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 comfort e Can be encryptd from a series of factorials start from one (1)This is e = 1 + 1/1 + +1/3 + and from this, the value of e is approximately 2.71828182845904. C urrently, when Mathematicians calculate the natural 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 cause multiplication and division easier especially when using very big poem, very depleted numbers and those with decimal points. Scientists use of the initiatory and 2nd 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(prenominal) calculations much quicker and less laborious. Although logarithm add-in are slowly decorous obsolete due to the invention of calculators and computers, logarithms themselves are still very useful. However, for manual calculations which likewise require a great horizontal surface 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 labor intensive and much faster the use of logarithms also increases the trueness of the results of calculations. This is because the use of logarithms allows minimal errors as the value in the table are approximations of the developed set 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 bracing technique is evidenced by the organic evolution of logarithmic methods based on logarithmic ordered seriess enables multiplication to be quick and lite since there is decreased long multiplication.Logarithms are ve ry essential in the bailiwick of astronomists, navigators, mathematicians and all other scientific fields like 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 example the measure of acidity of a nub is made easier when using logarithms. In the PH outgo substances have PH ranging from 0 7. A juice with PH of 4 is 10 times more bitter 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 capacity of the acidity changes towards the negative direction that is the higher(prenominal) the PH, the less acidic the solution.This was calculated by use of very dainty numbers such as 0.00001 that is pen in logarithmic form as (1 x 10-5) where 5 is the logarithm of the number (Standler B.R.1990). As we all know acidic solutions contain hydrogen ions H +(aq) and the pH is set in motion by measuring the logarithm of the slow-wittedness of these ions and since many people would be split by negative numbers, the PH is written assuming the negative sign 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 near pH number not by 1unit but by factor 10.electric and Electronic engineers use decibels and bels as units of measurements. The tam-tam is devised in a convenient substance to measure power loss in a telephone system wire rather than giving in amplifiers originally, the bel used to represent the amount of signaling power loss due to apology over a standard continuance of electrical cable, however, it is presently defined in terms of logarithms of base 10. The Richter scale that is used to measure the earthquake intensity is a perfect analogy of the bel scale. The 6.0 Richter earthquakes are 10 times more stringy than a 5.0 Richter earthquake . This means that an advantage of using a logarithmic measurement scale is the tremendous range of extension low-priced by a relatively small span of numerous values.ReferenceStrandler, R.B 1990 editorial Mathematics for engineers. The journal ofUndergraduate mathematics and its application vol II, pages 1-6, springOlds, C, D, 1963. Continued fractions, ergodic House New YorkWeisstein, Eric W. Natural logarithm from math world a due west web resourceAccessed online on 23/09/07