The concept of the log was invented in the 17th century to speed up calculations. The use of logs reduced the time needed for finding products of large numbers. The log was used as the basis in numerical calculations for more than 300 years until the perfection of mechanical calculating machines in the late 19th century and computers in the 20th century rendered them obsolete for large-scale computations. As far as any entrance exam is concerned, the log is an important topic.
The scientists quickly adopted the method of the log since the properties were used to simplify difficult and long calculations. They could also use the log to find the product of two numbers by checking each number’s log in a special table. We represent the natural log as ln n ( base e = 2.71828). It is one of the useful functions in Maths, which has applications in mathematical models. The logarithm (log) is the power to which a base must be raised to yield a given number. If x is the log of n to the base b, we can represent this as x = logb n. For example 102 = 100. Hence 2 is the log of 100 to the base 10. Similarly 24 = 16. So 4 is the log of 16 to the base 2. We can also write it as 4 = log216.
Laws of Logarithm
(1) Product law: logx (ab) = logx a + logx b
To find the product of two numbers, we find the log of each number from a table, then add the log together. Then check the table to find the number with that obtained logarithm. It is called the antilogarithm. For example, if we have to find 1000×100, finding the log of 100 (2) and log of 1000 (3). Then add the log, we get 2+3 = 5, and then find the antilogarithm of 5. We get 100,000.
(2) Ratio: logx (a/b) = logx a – logx b
To find the ratio of two numbers, we find the log of each number from a table, then subtract the log. Then check the table to find the number with that obtained log.
(3) Power: logx ab = b logx a
(4) Change of bases: logx a = logb a logx b
Central Limit Theorem (CLT)
CLT is a statistical theory which states that when the large sample size has a finite variance, the samples will be normally distributed. According to the central limit theorem, whenever a random sample of size n is taken from any distribution with mean and variance, then the sample mean will be approximately normally distributed with mean and variance. The higher the value of the sample size, the better the approximation to the normal. We can apply this theorem to almost all types of probability distributions.
There are some assumptions for the CLT theorem. The size of the sample should be large.
It should be drawn randomly following the condition of randomization. Also, the samples should not depend on each other. It should not influence the other samples. When the sampling is done without replacement, the sample size shouldn’t exceed 10% of the total population.