The Natural Logarithm

The function $\ln x = \int_{1}^{x} \frac{d t}{t}$ , the product law that follows from that area rather than being assumed, the number $e$ defined as the point where the area reaches 1, and the exponential recovered as its inverse.

Definition

Definition by area. For $x > 0$ ,

\ln x = \int_{1}^{x} \frac{d t}{t} .

For $x > 1$ this is the area under $1 / t$ from $1$ to $x$ ; for $0 < x < 1$ the orientation reverses and the value is negative; and $\ln 1 = 0$ because the interval is empty. The integrand $1 / t$ is continuous on $(0, \infty)$ , so the integral exists for every positive $x$ . It is undefined at $x \leq 0$ , since the path would cross the singularity at $t = 0$ .

The derivative. By the fundamental theorem of calculus, differentiating an integral with respect to its upper limit returns the integrand:

\frac{d}{d x} \ln x = \frac{1}{x}, x > 0 .

So $\ln$ is the antiderivative of $1 / x$ , the power whose integral the power rule cannot produce.

The laws, as consequences.

Law	Holds for
$\ln (a b) = \ln a + \ln b$	$a, b > 0$
$\ln (a / b) = \ln a - \ln b$	$a, b > 0$
$\ln (a^{r}) = r \ln a$	$a > 0$ , $r$ real

Each is derived from the definition, not assumed, and together they are why logarithms convert multiplication into addition.

The number $e$ . Since $\ln$ is strictly increasing, its derivative $1 / x$ being positive, and takes arbitrarily large and arbitrarily negative values, it attains the value 1 exactly once. That input is the definition of $e$ :

\ln e = 1, that is \int_{1}^{e} \frac{d t}{t} = 1 .

Numerically $e = 2.718281828459$ .

The exponential as inverse. Being strictly increasing, $\ln$ is injective, with range all of $R$ ; its inverse is $\exp$ , written $e^{x}$ , so that

\ln (e^{x}) = x for all real x, e^{\ln x} = x for x > 0 .

Other bases. For $b > 0$ , $b \neq 1$ ,

\log_{b} x = \frac{\ln x}{\ln b},

so every logarithm is a constant multiple of this one.

Growth. $\ln x \to \infty$ as $x \to \infty$ , but slower than any positive power of $x$ : $\ln (10^{100}) = 230.258509$ , and $\ln x$ exceeds 100 only beyond $x = e^{100} \approx 2.688 \times 10^{43}$ .

Assumptions and scope

$\ln x$ is defined only for $x > 0$ . The integral from 1 would otherwise cross the singularity of $1 / t$ at $t = 0$ .
The antiderivative of $1 / x$ on the negative axis is $\ln | x |$ , valid on any interval not containing 0; an integral spanning 0 does not exist.
The laws require positive arguments. $\ln (a b) = \ln a + \ln b$ needs $a, b > 0$ individually, not merely $a b > 0$ , which is why combining logarithms can enlarge a domain and introduce extraneous roots.
There is no law for $\ln (a + b)$ . Logarithms convert products to sums, never sums to anything simpler.
The series $\ln (1 + u) = u - \frac{u^{2}}{2} + \frac{u^{3}}{3} - \dots$ converges only for $- 1 < u \leq 1$ , and at $u = 1$ so slowly as to be useless for computation.
$\ln$ is strictly increasing and unbounded above, yet grows more slowly than any positive power of $x$ : unboundedness and slow growth are compatible.

Worked material

Example

Areas, laws and a series, computed

1. The definition, evaluated numerically. Computing $\int_{1}^{x} \frac{d t}{t}$ by Simpson's rule and comparing with the known logarithm:

$x$	Area $\int_{1}^{x} d t / t$	$\ln x$	Difference
$1$	$0.000000000000$	$0.000000000000$	$0$
$2$	$0.693147180560$	$0.693147180560$	$2.2 \times 10^{- 16}$
$e$	$1.000000000000$	$1.000000000000$	$1.1 \times 10^{- 16}$
$3$	$1.098612288668$	$1.098612288668$	$5.1 \times 10^{- 15}$
$4$	$1.386294361120$	$1.386294361120$	$5.8 \times 10^{- 15}$
$8$	$2.079441541680$	$2.079441541680$	$1.8 \times 10^{- 15}$
$0.5$	$- 0.693147180560$	$- 0.693147180560$	$2.2 \times 10^{- 16}$

The definition is not a formal gesture: it computes. Note the row at $x = 0.5$ , where the area is negative because the limits run backwards, and the row at $x = e$ , where the area is exactly 1, that row is the definition of $e$ .

2. The laws, checked. $\ln 4 = 1.386294361120$ and $\ln 2 + \ln 2 = 1.386294361120$ ; $\ln 35 = 3.555348061489$ and $\ln 5 + \ln 7 = 3.555348061489$ ; $\ln 4 = 1.386294361120$ and $\ln 0.5 + \ln 8 = 1.386294361120$ , this last mixing a negative logarithm with a positive one.

The power law across several exponents:

$n$	$\ln (2^{n})$	$n \ln 2$
$1$	$0.693147180560$	$0.693147180560$
$2$	$1.386294361120$	$1.386294361120$
$4$	$2.772588722240$	$2.772588722240$
$8$	$5.545177444480$	$5.545177444480$
$10$	$6.931471805599$	$6.931471805599$

This is what "turns multiplication into addition" means concretely: computing $\ln (2^{10})$ needs one multiplication rather than ten.

3. Change of base. $\log_{10} 1000 = 3$ , $\log_{2} 1024 = 10$ , $\log_{3} 81 = 4$ . Each equal to $\ln x / \ln b$ to twelve decimals. Every logarithm is this one, rescaled.

4. The series, and where it fails. For $| u | < 1$ ,

\ln (1 + u) = u - \frac{u^{2}}{2} + \frac{u^{3}}{3} - \frac{u^{4}}{4} + \dots

At $u = 0.5$ , converging to $\ln 1.5 = 0.405465108108$ :

Terms	Value	Error
4	$0.401041666667$	$4.42 \times 10^{- 3}$
8	$0.405315290179$	$1.50 \times 10^{- 4}$
16	$0.405464803171$	$3.05 \times 10^{- 7}$
32	$0.405465108106$	$2.38 \times 10^{- 12}$

At $u = 0.1$ , converging to $\ln 1.1 = 0.095310179804$ , just 8 terms give $0.095310179702$ , error $1.0 \times 10^{- 10}$ , and 16 terms reach the limit of double precision. Smaller $u$ converges dramatically faster.

At the boundary $u = 1$ it becomes the alternating harmonic series, converging to $\ln 2$ so slowly as to be useless:

Terms	Value	Error
$10$	$0.6456349206$	$4.75 \times 10^{- 2}$
$100$	$0.6881721793$	$4.98 \times 10^{- 3}$
$1000$	$0.6926474306$	$5.00 \times 10^{- 4}$
$10000$	$0.6930971831$	$5.00 \times 10^{- 5}$

Ten thousand terms and still wrong in the fifth decimal. Each tenfold increase in work buys one digit. Compare 32 terms at $u = 0.5$ giving twelve correct digits. The contrast is the interval of convergence at work: the series converges for $- 1 < u \leq 1$ , but at the endpoint only barely, which is why practical computation reduces the argument with the laws first rather than using the series directly.

5. Slow growth. $\ln 10 = 2.302585$ , $\ln 10^{6} = 13.815511$ , $\ln 10^{12} = 27.631021$ , $\ln 10^{100} = 230.258509$ . Each factor of $10^{6}$ adds about $13.8$ . The function is unbounded, yet reaching 100 requires $x > 2.688 \times 10^{43}$ .

Common errors

Common misconception

The logarithm distributes over addition, so $\ln (a + b) = \ln a + \ln b$ .

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Techniques of Integration (used by)
Power Series, Taylor Expansion and the Remainder (related)

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The Natural Logarithm

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Calculus: Early Transcendentals (2020)

The Natural Logarithm

Assumptions and scope

Worked material

Areas, laws and a series, computed

Common errors

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