The Logarithm of a number is the exponent by which another fixed value the base has to be raised to produce that number. For example the logarithm of 1000 to base 10 is 3as 1000=103. Logarithms were introduced by John Napier in the early 17th century for simplification of calculation and were widely adopted by scientists, engineers and others to perform computations more easily using logarithm tables. The logarithm to base b=10 is called the common logarithm and has lot of applications in science and engineering, while the natural logarithm has the constant e (2.718281828) as its base and is written as ln(x)or loge(x).
This common log is used in most of exponential scales (such as 23) in chemistry such as pH scale (for measurement of acidity and alkalinity), Richter scale (for measurement of intensity of earthquakes), and so on. It is so common that if you find no base written, you can assume it to be log x or common log.
Natural logarithm is widely used in pure mathematics specially calculus. The natural logarithm of a number x is the power to which e have to be raised to equal x. For example, ln(7.389…) is 2, because e2=7.389. The natural log of e itself (ln(e)) is 1because e1=e, while the natural logarithm of 1 (ln(1)) is 0, since e0=1.
The question is “the reason of choosing 10 is obvious, but why e=2.718…”?
The answer is that it back to 300 years or more ago to Euler (which e comes from his name). The function ex is the only function that its derivative (and consequently its integral) is itself. ( ex’ = ex ), no other function has this characteristic. The number e could be achieved by several numerical and analytical methods, more often infinite summations. This number has more important rule in complex analysis.
Suppose you have a hundred rupees, and the interest rate is 10%, you will have Rs. 110, and the next time another 10% of Rs. 110, will raise you amount to Rs. 121, and so on… What happens when the interest is being computed continuously (all the time)? You might think you would soon have an infinite amount of money, but actually, you have your initial deposit times e to the power of the interest rate times the amount of time:
where k is growth rate or interest rate and t is time period, P is Value at time t and P0 is Value it time t=0.
Intuitive explanation is: ex is the amount of continuous growth after a certain amount of time. The natural log gives you the time needed to reach a certain level of growth. That is, ex is the amount of continuous growth after a certain amount of time and natural log is the amount of time needed to reach a certain level of continuous growth.