Logarithmic functions

John Napier is the man credited to have contributed hugely to the fields of science, philosophy and mathematics. Many believe that he is the brainchild of the modern computer science since he helped in making multiplication, division and root extraction much easier especially for very large numbers. In the world of mathematics the genius of a man, John Napier is credited to have invented the logarithms as early as 1614 and states in his book The “Descriptio” that he started contemplating the idea of logarithms twenty years earlier which was in the year 1594. Using Napier’s table in his book, calculations were made using the logarithm identities. These are the present first and second laws of logarithms: Log XY = Log X + Log Y as well as

Log X / Y =Log X – Log Y. In his book “the Descriptio”John Napier defined logarithmic function as a differential equation.

When the base is “b” and the variable is “x” the logarithm to the base “b” of the variable “x” can be defined as the “power to which you would raise “b” to get “x”. Other scientists define logarithm as “the exponent to which the base must 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 are three laws of logarithm that scientists use in interpreting logarithm: These laws are:

The product to sum rule – This law expresses that the product of a logarithm is equal to the sum of the individual 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 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 X

Common logarithms

As earlier identified a logarithm to be valid 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 indicated 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 more prevalent when using arithmetic series as opposed to geometric series.

Natural logarithms

In 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 great mathematician from Switzerland by the name Leonhard Euler. Currently “e” is the base used in calculus and has since been named as “natural base”. The value “e” Can be calculated from a series of factorials starting from one (1)

This is; “e” = 1 + 1/1 + ½ +1/3 + ¼… and from this, the value of “e” is approximately 2.71828182845904. Currently, 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 make multiplication and division easier especially when using very big numbers, very small numbers and those with decimal points. Scientists use of the 1st 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 calculations much faster and less laborious. Although logarithm table are slowly becoming obsolete due to the invention of calculators and computers, logarithms themselves are still very useful. However, for manual calculations which also require 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 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 errors as the values 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 Napier’s 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 quick and easy since there is decreased long multiplication.

Logarithms are very essential in the work 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 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 know acidic solutions contain hydrogen ions H+(aq) and the pH is found by measuring the logarithm of the concentration of these ions and since many people would be confused 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 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 convenient way to measure power loss in a telephone system wiring rather than giving in amplifiers – originally, the bel used to represent the amount of signal power loss due to resistance 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 earthquake 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 tremendous range of extension affordable by a relatively small span of numerous values.

Reference:

Strandler, R.B 1990 “Editorial”: Mathematics for engineers. The journal of

Undergraduate mathematics and its application vol II, pages 1-6, spring

Olds, C, D, 1963. Continued fractions, Random House New York

Weisstein, Eric W. “Natural logarithm” from math world a wolfram web resource

Accessed online on 23/09/07