The Natural Logarithm and the Number e

What is $e$?

Euler's number $e \approx 2.71828...$ is one of the most important constants in mathematics. It's defined as:

$e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$

Step-by-step: watching convergence

Compute $\left(1 + \frac{1}{n}\right)^n$ for increasing $n$:

• $n = 1$: $(1 + 1)^1 = 2.000$
• $n = 2$: $(1.5)^2 = 2.250$
• $n = 5$: $(1.2)^5 = 2.488$
• $n = 10$: $(1.1)^{10} = 2.594$
• $n = 100$: $(1.01)^{100} = 2.705$
• $n = 1000$: $(1.001)^{1000} = 2.717$
• $n = 10000$: $(1.0001)^{10000} = 2.71815$

The values settle down to $e = 2.71828...$

Where $e$ comes from: compound interest

If you invest $1 at 100% annual interest compounded $n$ times per year, after 1 year you have $(1 + 1/n)^n$ dollars. With continuous compounding ($n \to \infty$), you get exactly $e$ dollars. This is why $e$ appears in finance, biology (population growth), physics (radioactive decay), and everywhere exponential change occurs.

The natural logarithm

$\ln x = \log_e x$ is the logarithm with base $e$. It's "natural" because $\frac{d}{dx}[\ln x] = \frac{1}{x}$ — the simplest possible derivative for a log function.

Key values: $\ln 1 = 0$, $\ln e = 1$, $\ln e^2 = 2$.

Try it in the visualization

Watch $(1 + 1/n)^n$ converge as $n$ increases. The graph shows the value approaching the $e$ reference line. A table displays the sequence of values.