Logarithms (math)


Logarithms

 

In algebra topic of math subject we denote expressions as some functions of the variable. For example,the expression ax2 + bx + c may be denoted as f(x), meaning a ‘function of x’. There are several types offunctions depending upon the concepts used in the relation of the functions. A logarithmic function is oneof such types where the concept of logarithms is used.



To define what a logarithm is let us start with simple power expressions. We know that 10 3 = 1000.That is when 3 is the exponent for a base 10, the value of the expression is 1000. We can say this inanother way that 1000 is obtained if the exponent is 3 for a base of 10. In other words, we can callthat ‘something’ of 1000 for a base 10 is 3. That’ something is defined as ‘logarithm’. Therefore, thelogarithm of a number x is defined as ‘a’ which could be the exponent of a given base ‘b’. Algebraically itis written as, logb(x) = a. It spontaneously means thatba = x.


Laws of Logarithms

Logarithms of same base follow certain laws while operating with them. These laws are simple derivationsfrom laws of exponents and greatly help in solving logarithmic problems. Suppose there are two

logarithms say, logb(x) and logb(y), then

logb(x) + logb(y) = logb(x*y)

logb(x) - logb(y) = logb(x/y)

logb(xn) = n*logb(x)

Another important law is change of base. It helps to find the value of the logarithm to an unknown base ifthe logarithm to another base is known. It states as,logp(x) = [logb(x)]/[logb(p)]

Since no exponent to any base can make the value of a power expression as, 0 or negative, logarithm of0 and logarithm of a negative are not defined.Let us see a few examples of how these rules help us in solving logarithms. We know 101 = 10.Therefore, what follows is log10(10) = 1. The logarithm of 100 is found as log10(100) = log10(102)= 2*log10(10) = 2*1 = 2. Similarly, log10(30) is simplified as log10(30) = log10(10*3) = log10(10) +log10(3) = 1 + log10(3)


Logarithm Examples

Logarithms are used in many applications. The measures of intensity of earthquakes, the amount ofhydrogen ion concentration are a few examples where logarithm to base 10 is used. The logarithm tobase 10 is called as ‘common logarithm’ and denoted simply as ‘log’, without mentioning 10 as the base.Another very important example of logarithm is ‘natural logarithm’ where the exponential constant ‘e’ isused as the base. It is denoted as ‘ln’.