The Natural Base e and the e-Function

The number e ≈ 2.71828 is the base of natural exponential growth, and the e-function f(x) = e^x is the exponential function whose slope at every point equals its value. In brief: e is the limit of (1 + 1/n)ⁿ as n grows without bound, and f(x) = e^x has the special property that its derivative is e^x itself. Continuous growth at rate r gives A = A₀e^rt.

Among all exponential functions, the base e is the natural choice in mathematics and science, because it makes formulas simplest. The number appears in continuous compounding, in radioactive decay, in the normal distribution and in the complex numbers. This guide explains where e comes from, its properties, how to use e^x in growth problems and the relation to the natural logarithm.

What is the number e?

The number e is an irrational constant with decimal expansion 2.718281828459045… It can be defined in several equivalent ways:

• As a limit: e = lim (1 + 1/n)ⁿ as n → ∞.
• As an infinite sum: e = 1 + 1/1! + 1/2! + 1/3! + 1/4! + …
• As the unique base for which the graph of b^x has slope 1 at x = 0.

The sum converges very fast: 1 + 1 + 0.5 + 0.1667 + 0.0417 + 0.0083 + 0.0014 = 2.7181 after only seven terms. The number is irrational, and in fact transcendental, as shown by Hermite in 1873.

How does e arise from compound interest?

Suppose 1 dollar is invested for one year at an interest rate of 100%. With compounding n times per year, the final amount is (1 + 1/n)ⁿ. Compounding once gives 2, twice gives 2.25, monthly gives 2.613, daily gives 2.7146 and, in the limit of continuous compounding, e ≈ 2.71828. The growth is bounded: more frequent compounding gives more, but never more than e.

For a general rate r and time t, continuous compounding gives A = P · e^rt.

What are the properties of the e-function?

• The domain is all real numbers, the range is y > 0 and the asymptote is y = 0.
• The graph passes through (0, 1) and (1, e) ≈ (1, 2.718).
• The function is increasing and the tangent line at (0, 1) has slope 1.
• The derivative of e^x is e^x, and the integral of e^x is e^x + C.
• The inverse is the natural logarithm: ln(e^x) = x and e^{ln x} = x for x > 0.
• Any exponential function can be written with base e: b^x = e^{x ln b}. For example 2^x = e^0.6931x.

How do you model continuous growth with e?

If a quantity changes at a rate proportional to its size, it follows N(t) = N₀e^kt. The constant k is the continuous growth rate (k > 0) or decay rate (k < 0). The relation to a yearly factor b is b = e^k.

Example 1: continuous interest

1,000 dollars at 4% continuously compounded for 5 years: A = 1000 · e^{0.04 · 5} = 1000 · e^0.2 ≈ 1000 · 1.2214 = 1,221.40 dollars. With yearly compounding it would be 1000 · 1.04⁵ ≈ 1,216.65 dollars.

Example 2: decay

A substance decays as N(t) = 200e^−0.1t grams. After 10 days: 200e⁻¹ ≈ 73.6 g. The half-life is t = ln 2/0.1 ≈ 6.93 days.

Example 3: solving for time

When does 500e^0.03t reach 1,500? e^0.03t = 3, so 0.03t = ln 3 ≈ 1.0986 and t ≈ 36.6.

Compounding table and values of eˣ

Where does e come from?

The number appeared in connection with compound interest in the work of Jacob Bernoulli at the end of the seventeenth century, who studied the limit of (1 + 1/n)ⁿ. The letter e was introduced by Leonhard Euler in the 1730s, and it is often said to stand for “exponential”, though Euler’s own reason is not documented. Napier’s work on logarithms earlier contained a table connected with the number, but not the constant itself.

Where is e used today?

The bell curve of the normal distribution contains e^−x²/2. Electrical engineers describe the charging of a capacitor by 1 − e^−t/RC. Euler’s formula e^iπ + 1 = 0 connects e with complex numbers and π. Banks, scientists and statisticians use e whenever a process changes continuously.

Which mistakes are common?

Treating e as a variable

The letter e is a fixed number, about 2.718, not an unknown.

Confusing e^x with x^e

In e^x the exponent varies. In x^e the base varies, and the function is a power function.

Rounding too early

Use the e key of a calculator and keep several decimals until the end.

Mixing yearly and continuous rates

A continuous rate of 5% corresponds to a yearly growth of e^0.05 − 1 ≈ 5.13%, not 5%.