Why Is the Natural Logarithm Base Called ‘e’? History and Proof Explained

Mathematical Definition of e

e is defined as the important limit \(\displaystyle e=\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n}\), which arises naturally in calculus and analysis.

Origin of the Symbol

There are three main explanations for why the letter e is used. The first attributes it to the Swiss mathematician Leonhard Euler, who employed the symbol in 1727, taking the first letter of his name. The second links it to the English word “exponential,” whose initial letter is also e. A third, less documented, suggestion is that e was simply the next available letter after other symbols were taken.

Why e Is Chosen as the Base of Natural Logarithm

When differentiating the logarithm function \(\log_{a}x\), the derivative simplifies to \(\frac{1}{x\ln a}\). Choosing a = e makes \(\ln a = 1\), so the derivative of \(\ln x\) is simply \(\frac{1}{x}\). Any other base would introduce an extra constant factor, making formulas less elegant.

The symbol "e" for the natural logarithm comes from the abbreviation of the word “exponential,” whose first three letters are "exp," and the first letters of “logarithm” (l) and “nature” (n) combine to form the notation.

Monotone Bounded Theorem

Monotone bounded sequences have limits.

This theorem, fundamental in calculus, states that a sequence that is monotonic (either non‑decreasing or non‑increasing) and bounded must converge to a limit.

Proof of the Existence of e

Consider the sequence \(a_n = \left(1+\frac{1}{n}\right)^{n}\).

(1) The sequence is monotone increasing. By expanding the binomial expression, each term adds a positive amount, so \(a_{n+1} > a_n\).

(2) The sequence is bounded above. Using the inequality \(1+\frac{1}{n} < e^{1/n}\) and properties of exponentials, one can show \(a_n < 3\) for all n, providing an upper bound.

Since the sequence is both monotone increasing and bounded, the Monotone Bounded Theorem guarantees that it converges. Its limit is defined as the constant e.

Compound Interest Problem

Let P be the principal and r the annual interest rate. After t years with continuous compounding, the balance is \(P\,e^{rt}\). If interest is compounded n times per year, the balance becomes \(P\left(1+\frac{r}{n}\right)^{nt}\). As n → ∞, this expression approaches the continuous‑compounding formula, illustrating why e appears naturally in finance.

References

段美松. 浅谈自然对数的来源[J]. 新智慧, 2018(34):17.

柴英明. 自然对数的底 e 与年化利率[J]. 中外企业家, 2016, No.532(14):241‑242.