The Number e and the Natural Logarithm (ln) Explained

Q: What is e in maths?

e 2.71828 is the limit of (1 + 1/n)^n as n grows. It is the base for which the exponential function's slope equals its own value.

Q: What is the difference between log and ln?

log x usually means base 10, while ln x means base e. They obey the same laws and differ only by a constant factor.

Where $e$ comes from, why calculus prefers $e^x$ over every other exponential, what $\ln$ means, how to use it to solve equations, and the continuous growth formula $A = Pe^{rt}$ .

3 min read Supports 4 lessons

Put £1 in a fantastically generous account paying $100\%$ interest a year. Added once at the end of the year, you finish with £2. But what if the interest is split into smaller pieces and added more often?

$\left(1 + \tfrac{1}{1}\right)^{1} = 2$
Once a year.

1/5

Euler's number

e = \lim_{n\to\infty}\left(1 + \frac{1}{n}\right)^n \approx 2.71828

Like $\pi$ , $e$ is irrational: its decimals go on for ever without repeating. It is named after the mathematician Leonhard Euler.

Why calculus prefers e

Every exponential $b^x$ grows at a rate proportional to its own size. For base $e$ , and only base $e$ , that proportion is exactly $1$ : the slope of $y = e^x$ at every point equals its height.

Try $b \approx 2.72$ . At $(0, 1)$ the curve's slope is $1$ , matching its height.

Key idea

Because $e^x$ is its own rate of change, it is the natural base for anything that grows or decays in proportion to its size: populations, investments, radioactive material, cooling.

The natural logarithm

The logarithm with base $e$ is called the natural logarithm, written $\ln x$ . It answers: $e$ to what [[exponents|power]] gives $x$?

Natural logarithm

\ln x = \log_e x \qquad \ln e = 1, \quad \ln 1 = 0, \quad e^{\ln x} = x, \quad \ln(e^x) = x

Every log law works for $\ln$ : $\ln(ab) = \ln a + \ln b$ and $\ln(a^n) = n\ln a$ .

Worked example

Solve $e^{2x} = 10$ .

1
Take $\ln$ of both sides.
$2x = \ln 10$

Continuous growth

When growth happens all the time rather than in steps, the formula uses $e$ directly.

Continuous growth

A = Pe^{rt}

$P$ is the starting amount, $r$ the continuous growth rate and $t$ the time.

Worked example

£1000 grows continuously at $5\%$ a year. How long until it doubles?

1
Set up the equation.
$2000 = 1000e^{0.05t}$

Tip

You can type exact answers like `ln(10)/2` into the answer box in the course.

Common questions

What is e in maths?: $e \approx 2.71828$ is the limit of $(1 + \frac{1}{n})^n$ as $n$ grows. It is the base for which the exponential function's slope equals its own value.
What is the difference between log and ln?: $\log x$ usually means base $10$ , while $\ln x$ means base $e$ . They obey the same laws and differ only by a constant factor.

Practise the number e

The The Number e & Natural Logarithms lesson has worked examples and unlimited practice that adapts to you.

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Common questions

What is e in maths?

$e \approx 2.71828$ is the limit of $(1 + \frac{1}{n})^n$ as $n$ grows. It is the base for which the exponential function's slope equals its own value.

What is the difference between log and ln?

$\log x$ usually means base $10$ , while $\ln x$ means base $e$ . They obey the same laws and differ only by a constant factor.