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    Even More About e, the base of the Natural Logs


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Facts
  • The number e is the base of John Napier's natural logarithms created in the 1600s.   The symbols ln(x) mean loge(x).
        To experiment with log computation visit How To Appreciate Logs Without Feeling Out On a Limb.
     
  • It is named for Leonhard Euler who studied it in the 1700s.
     
  • It is named for Leonhard Euler who studied it in the 1700s.
     
  • It is irrational, never a fraction. It is a non-repeating, non-terminating decimal.

     
  • It is transendental (is NOT a root of a non-zero polynomial equation with rational coefficients).

     
  • The first few digits are: 2.718281828459045235360. There is even a little story to help one memorize the first few digits.
     
  • When used as the base of an exponential or power function, the function's slope works beautifully with the function 1/x, but the value of e is needed. So, "Solve" for e.

     
  • It is defined, by Jacob Bernoulli in the 1600s, to be the limit, as n goes to infinity, of the nth power of the sum of 1 and the reciprocal of n. Experiment with this idea by letting t, P0, and r equal 1 and changing n. The larger the n, the closer the limit is to e.

     
  • It may be approximated with this infinite sum. The larger the n, the closer the sum is to e.

     
  • When used as the base of an exponential or power function, cx where c > 0 and c 1, it becomes ex.   See An Exponential or Power Function.
     
  • If e is the base of a power function, the function, its slope function (derivative), and its area function (integral) all the same. In "Parent Functions, Their Slope Functions, and Area Functions" see The Exponential Function, exp(x) or ex

     
  • If e is the base of the log, ie. it is ln(x), the derivative is the reciprocal function. The antiderivative of the reciprocal, x>0, is the natural log.

     
  • It is used when modeling continuous growth and decay. Experiment with it.

     
  • It is also used in probability to write the normal distribution, and and trig to write hyperbolic functions,

     
Examine the normal, logistic, and exponential functions with a spread sheet.


A Story to Spell Out Digits of e

    The approximation 2. 7 1828 1828 45 90 45 2 35 3 6 0 is a mouthful. Perhaps this will help.

There are
2 things you should know about Andrew Jackson.
He was the
7th pesident of the United States.
I'd like to say his term of office began in
1828 but that's not true. It ran form 1829 to 1837.
I'd REALLY like to say his term of office began in
1828 but that's not true. It ran form 1829 to 1837.
You should also know the angles created when a diagonal is drawn in a square. They are in this order for this purpose:
45
90
45
The
2nd US presdent was John Adams, who was born in the 1700s in
'35.
He beat Jefferson by
3 votes: 71 to 68. The next election Jefferson beat Adams 73 to 65.
He really did have
6 kids: Abigail, John Quincy (the US president),
Susanna, Charles, Thomas, Caroline
Of course there are more digits to e, but we've nothing to add to our story.
0


In Closing ...

    Hope you learned something and had some fun.

    Don't forget about:

A Program which Permits One to "Solve" for e.
Function and Relation Library.
Trig Functions Library
Exponential or Power Functions
A Bit about e
Linear vs Exponential Growth
Parent Functions, Their Slope Functions, and Area Functions


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