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Natural logs and Euler's Number

The Natural Exponential Function

Euler's Number

$$e=∑↙{n=0}↖{∞}1/{n!}$$

Euler's Identity

$$e^{iπ}+1=0$$

Euler's Equation

$e^{ix}= $cos$x + i$sin$x$

The irrational number $e$ is a fundamental natural number of great importance in mathematics. It was searched for by members of the Swiss Bernoulli family in the 17th century, but was finally defined accurately by Leonard Euler. For this reason, it is called Euler's number.

Mathematics log and e

e is irrational, and is defined by a series. In decimals, its value to 10 significant figures is e = 2.718281828..

log$_{10}x$ expresses the value of the exponent of 10 which will equal x. e.g. log$_{10}100 = 2$, log$_{10}1000 = 3$

The 'natural log' is log$_{e}x$. It is equal to the exponent of e which will equal x. A short-hand way to write log$_{e}x$ is lnx.

e.g. ln$e =1$, ln$1=0$, ln$7.39=2$, since $e^2=7.39$.

There is no exponent of e which will result in a negative number, so the domain of ln$x$ is $x>0$. the range is ℝ.

The infinite series

$$f(x)= ∑↙{n=0}↖{∞} {x^n}/{n!}= 1+x+{x^2}/2+ {x^3}/{2⋅3} + {x^4}/{2⋅3⋅4}+ ...... = e^x$$

Inverse function of the natural log

The inverse of $f(x)=e^x$ is $f^{-1}(x)=$log$_{e}x$

e.g. If $f(x) = e^{x-2}$:

For $f^{-1}(x)$, $x=e^{y-2}$

$y-3= $ln$x$, or $y=3+ $ln$x$

Compound Interest

Interest from invested capital is calculated from the formula:

$$A=C(1+r/n)^{nt}$$

where $A$ is the final amount (capital + interest), $C$ is the capital, and $r$ is the interest rate. $n$ is the number of times the interest is compounded (calculated and added to the capital) in a year, and $t$ is the number of years the money is invested for.

The more often the interest is compounded, the faster the capital grows. What is the limit as n approaches the largest number of compoundings possible? In other words, what would be the yield if the interest is compounded continuously and not at intervals?

Solution: For r = 1.0, ${lim}↙{n→∞} C(1+r/n)^{nt} = Ce$

e = 2.718281828...

This is Euler's number, and is an irrational number.

$a^x = e^{xlna}$

Exponential Growth

The probability that of a group of people selecting their hats after a meeting at random, no one selects their own hat, is $1/e$!

The reason that $e$ is so important is that nature obliges its growths and decays to use $e$, rather than the human inventions deriving from base-10 finger-counting systems.

Examples of nature's enthusiasm for Herr Euler's number are: micro-organism population growth rates, virus epidemics, nuclear chain reactions, heat transfer. Even human systems comply to $e$: compound interest, computer processing power, human population growth, and internet traffic development, can all be modelled on the natural logarithm and $e$.

Population Growth

What do a city and a petri dish of bacteria have in common?

Populations of every type follow an exponential growth function. The parameters, like fertility rate, number of children per family, etc., may vary from city to city around the world, but the curve has the same basic shape: an exponential curve.

Example: city life

Tokyo lights

The population of a city, in millions, is a function of time, t, in years:

$$P(t) = e^{(0.025)t}$$

Taking 2000 as year 0, $P(0) = 1,000,000$

At the end of one year, the population was: $P(1) = e^{(0.025)1} = 1.025315$ million, indicating an increase of 25,315 people.

In 2015, the population should be: $P(t) = e^{(0.025)×15} = 1.454991$, indicating a cumulative growth of 454,991 people in the 15 years since 2000.

When will the population reach 2 million?

Now, we need to solve for $t$:

$2 = e^{(0.025)t}$

$ln2 = lne^{(0.025)t}$

$ln2 = 0.025t$

$t = {ln2}/{0.025} = 28$ years, or 2028

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