Euler’s Number and Natural Logarithms 🤔

Author

The majority of the text and code has been generated by AI.
Editor: Witek ten Hove

Introduction

Euler’s number, often denoted as $e$, is an irrational number roughly equal to 2.71828. It might sound abstract, but it plays a vital role in mathematics, particularly when dealing with exponential growth or decay, and in concepts of continuous compounding in finance.

The natural logarithm, usually represented as $\ln$, is the logarithm to the base $e$. It’s fundamental in various fields, from solving exponential equations to analyzing exponential growth or decay patterns in data science.

Euler’s Number ($e$)

Euler’s number arises naturally in many areas of mathematics. For instance, consider money that is being continuously compounded. If you were to compound $1 at 100% interest over a year, but compound more frequently (daily, hourly, every second, and so on), the amount you’d have with continuous compounding is $e$ dollars.

Natural Logarithm ($\ln$)

The natural logarithm is the inverse operation to exponentiation with base $e$. In other words, if $y = e^x$, then $x = \ln(y)$.

Why are they Important?

Continuous Growth/Decay: Exponential growth or decay that happens continuously (like population growth or radioactive decay) can be described using Euler’s number.
Finance: In business, the concept of continuous compounding is rooted in $e$.
Data Analysis: The natural logarithm helps in linearizing exponential growth data, making it easier to analyze and model.

Code Example in Python

Let’s look at a real-world scenario:

Imagine a startup company’s user base grows continuously. In its initial stages, the number of users doubled every year. We can model this growth with the equation: $N(t) = N_0 \times e^{kt}$ Where:

$N(t)$ is the number of users after time $t$.
$N_0$ is the initial number of users.
$k$ is the growth constant.

Given that the users doubled every year, we can determine $k$.

Code

import numpy as np

# Given values
N_0 = 100  # Initial number of users
N_t = 200  # Number of users after 1 year

# Finding k using natural logarithm
k = np.log(N_t/N_0)

print(f"With growth constant k = {k} the number of users after 2 years = 100*e^(k*2) = {100*np.exp(k*2)}")

With growth constant k = 0.6931471805599453 the number of users after 2 years = 100*e^(k*2) = 400.0

Using this growth constant, a company can predict future user numbers and strategize its business operations accordingly.

Assignment

Basic: Determine the derivative of $f(x) = e^x$ using Python (see section on derivatives). Evaluate the derivative function $f'(x)$ at $x=1, 2\ and\ 3$ and compare with the corresponding values of $f(x)$. Explain in a short video what you’ve noticed and elaborate.

Stretch: Expand your video with a presentation of the most beautiful equation in mathematics.

Challenge: Share your content online (e.g. Linkedin or Medium), gather feedback and write a reflection on it.