Natural Logarithm and “e” = 2.71828……(Continued)

In previous post i concluded with an open question of “integration of x-1” and if you were aware of calculus formula you would have guessed right. This post revolves around a path to find the solution, leading to famous “e“.

While one part of mathematicians were puzzled about this question, there were other set of mathematicians working on differentiation of functions known to them. They were not worried much about this integration of x-1, they kept inventing new formulas for different functions. One such function was Exponential function.

Differentiation of Exponential function

Differentiation formula can be used to calculate the slope of any exponential function of form y = bx . Unlike power function which has variable in base and constant in power (Eg: y=x2), exponential function has constant in base and variable in power. Higher the value of base, faster the increase in “y”.

Exponential Curve
Derivative equation

There are two IMPORTANT inferences made from above equations, which will be referred later in this post (read twice if it necessitates):

  • Slope of bx is proportional to itself.
  • “c” is a constant which is Slope of bx at x=0 and its value depends on base “b”.

Inverse function

Park in mind those two inferences until inverse function is briefed. Mathematicians were aware of inverse functions and their relation with actual function using differentiation. Inverse functions are those, which will reverse the output back to its input. Eg: f(x) = y = 2x, inverse function will be g(y) = y/2 which will give back x. Relationship between derivative of a function and derivative of its inverse function can be found using chain rule of differentiation. In previous example dy/dx = 2; dx/dy = 1/2; Take any function and its inverse, their derivatives multiply to produce identity. In other words, slope of an inverse function is “-1 power of slope of its normal function”.

Inverse equations
Bonus to motivate you to find derivative of trigonometric inverse functions

Savior in finding “c”

Now going back to our exponential function. We differentiated bx and ended up in “c” which is the slope of function at x=0. Inverse function for bx is logb(y). If you are surprised on this sudden magic, substitute bx into logb(y) => logb(b)x = xlogb(b) = x;

Note: logb(b) = 1; logb(1) = 0;

Logarithm is another interesting topic which would need another separate post to cover. For now lets not deviate from our objective but remember the base of the logarithm “b”.

Logarithm and Exponential functions

In exponential world x = 0 gives y = b0 = 1; In logarithm to the base “b” world y = 1 gives x = 0; Value of x and y remains same in both worlds but the dependent variable will be x in first case and y in its inverse case.

c” in exponential world is slope of curve when x = 0; “1/c” will be the slope at y = 1 in logarithm to base “b” world; How?

We already seen the relationship between derivatives of normal and its inverse function, which will help to find the derivative of logb(y) easily without much computation.

Formula for differentiation

So slope of the exponential function at x = 0 is “-1 power of” slope of its inverse function at y=1;

Phew ! Now, do you remember the question asked in part 1? What is anti-derivative of x-1? We are now just one step ahead of meeting our boss “e“.

FINALLY We ARe HeRe !

Logb(y) gave derivation as 1/cy. Logb(x) will give derivation as 1/cx (Just a change in variable does not cost anything). In some way if we make c=1, then anti-derivative of 1/x can be found.

We have two final tasks now:

  • Derive formula for “c”.
  • Since “c” depends on base “b”, compute “b” to make “c” = 1.
Value of exponents
limit does not depend on base “b”, so log and limit is swapped

The value inside parenthesis should be constant since it involves only ∆x which tends to 0. Time to reveal the formula for “e“.

Value of exponents

e” was chosen by Swiss mathematician “Euler” and popularly called as euler constant.

First inference : "e" is a mathematical constant.
Value of exponents
Second inference: "e" is also the base of the logarithm in order to make "c" = 1. 

Logarithm to the basee” is known as Natural logarithm and most textbooks refer it as ln(x). This raises curtain for the question asked before. Anti-derivative of x-1 is ln(x).

Logarithm x differentiation
Third inference: In our exponential function bx, if b = e, then the derivative output will be itself
Value of exponents

Reason for the term “natural” is, exponential growth appears in most of natural occurring phenomenon around us. Population growth in any year depends on the total population in that year. A hot cup of coffee will cool down at a rate proportional to the temperature at that point.

After a small and quick change of variables, e can be updated as below:

Value of exponents
n1/n1 + 1/n(1 + 1/n)n
1122
20.51.52.25
100.11.12.593742
1000.011.012.704814
10000.0011.0012.716924
100000.00011.00012.718146
1000000.000011.000012.718268
10000000.0000011.0000012.718280
100000000.00000011.00000012.718281
Fourth inference: As "n" tends to higher value, e tends to its actual value. With above value of n= 10 million, e is 2.71828....

Finally, the job is done. Below mind map summarizes the path taken to reach “e“.

Derivation of e value

Before i close this post, i would like to put the general formula for differentiation of exponential function.

Exponent value summary

For the sake of brevity, we will always represent this number 2.71828… by the letter e. Now i will have less distraction.

– Leonhard Euler

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