Types of Logarithmic functions and application
Logarithmic functions are the inverses of exponential functions, and any exponential function can be expressed in logarithmic form. Similarly, all logarithmic functions can be rewritten in exponential form. Logarithms are really useful in permitting us to work with very large numbers while manipulating numbers of a much more manageable size. Logarithms were introduced by John Napier in 1614 as a means of simplifying calculations.
So what is a logarithm? A logarithm is an exponent or power to which a base must be raised to yield a given number.
Common Logarithmic Function
A logarithm with base 10 is a common logarithm. In our number system, there are ten bases and ten digits from 0 to 9, here the place value is determined by groups of ten. You can remember common logarithms with the one whose base is common as 10.
Natural Logarithmic Function
The natural logarithm is different. When the base of the common logarithm is 10 the base of a natural logarithm is number e. Although e represents a variable.it is a fixed irrational number that equals 2.718281828459. Sometimes e is also known as Euler’s number or Napier’s constant. The letter e is chosen to honour mathematician Leonhard Euler. e looks complicated but is rather an interesting number. The function f (x) = loge x has multiple applications in business, economics, and biology. Therefore, e is an important number.
Real-life application of logarithmic functions
The Magnitude of an earthquake
In 1935 Charles Ritcher defined the magnitude of an earthquake with the formula:
where I is the intensity of the earthquake measured by the amplitude of a seismometer taken 100 km from the epicentre and S is the intensity of a standard earthquake, which is defined with an amplitude of 1 micrometre or 10−4 cm.
This means that the magnitude of a standard earthquake is:
The largest earthquakes on record had a magnitude of 8.9 on the Ritcher scale. This would be equivalent to intensity of 800,000,000. This shows that the Ritcher scale allows us to obtain more manageable numbers.
Chemical systems known as buffer solutions or chemical buffers have the ability to adapt to small changes in acidity to maintain a range of pH values. Buffer solutions have a wide variety of applications from aquarium maintenance to regulating pH levels in the blood.
The equation Henderson- Hasselbalch is used to calculate the pH of a buffer solution. Hasselbalch was studying the carbon dioxide that dissolves in the blood and the model of the pH of the blood in this situation is
where x is the partial pressure of carbon dioxide in the arteries, measured in torr.
Measure of entropy
Another application of logarithmic functions is the entropy of information The entropy of information H, in bits, of a randomly generated password consisting of L characters is given by LN , where N is the number of possible symbols for each character in the password.