The area that names the number

The first essay in this ladder found $e$ by asking which exponential curve has slope equal to height, and answered by squeezing: base 2 has a slope too small, base 3 too large, and somewhere between them sits the base that works. It is a good definition and it has a defect, which is that it begins by assuming a family of curves $b^x$ and everything those curves need — what $2^{\sqrt2}$ means, why $b^x$ is differentiable at all.

There is a second definition, which assumes almost nothing and produces the same number.

The definition #

Define $L(t)$, for $t > 0$, as the area under $1/x$ between 1 and $t$ — counted as negative when $t < 1$, so that the region is traversed right to left.

That is a definition with nothing in it but a hyperbola and the ordinary business of adding up rectangles. No exponentials, no fractional powers, no limits of $(1 + 1/n)^n$. Just the area under one of the simplest curves there is.

The claim is that this innocent quantity is the logarithm, and that the number where it first reaches 1 is $e$.

Why the areas add #

Here is the whole content, and it is a scaling argument with no calculus in it.

Consider the region under $1/x$ from $a$ to $ab$. Squash it horizontally by a factor of $b$, so that it now runs from $1$ to $a$ — wait, that is the wrong way round; take it the useful way. Start with the region from $1$ to $b$ and stretch it horizontally by a factor of $a$. It now runs from $a$ to $ab$. Stretching multiplies every width by $a$, so it multiplies area by $a$.

But the stretched region is no longer under the curve; its top edge sits at height $1/x$ where $x$ was the old coordinate, which is $a/x$ in the new one. To put it back under $1/x$, squash it vertically by a factor of $a$ — which divides the area by $a$ again.

The two operations cancel exactly. The region under $1/x$ from $a$ to $ab$ has the same area as the region from $1$ to $b$.

Reading that as a statement about $L$:

$$L(ab) = L(a) + L(b),$$

because the area from 1 to $ab$ splits at $a$ into the area from 1 to $a$ plus the area from $a$ to $ab$, and the second piece has just been shown to equal $L(b)$.

A logarithm has been produced from a picture. The defining property of a logarithm — turning products into sums — has come out of the fact that the curve $1/x$ is unchanged by the operation stretch across and squash down by the same factor, and that operation preserves area. No other curve on the plane has that property, which is why no other curve gives a logarithm.

The one curve this works for #

It is worth asking why the argument singles out $1/x$, since the stretch-and-squash move can be tried on anything.

Take any curve $f$, stretch the region under it from 1 to $b$ horizontally by $a$, and squash vertically by $a$. The result is the region under the curve $x \mapsto f(x/a)/a$, running from $a$ to $ab$, with the same area as before. For the argument to close, that curve has to be $f$ again — so $f$ must satisfy $f(x/a)/a = f(x)$ for every $a$, which after putting $x = a$ says $f(a) = f(1)/a$.

So $1/x$, up to a constant multiple, is the only curve for which this works. The logarithm is not one of several functions that could have been produced this way; it is the unique one, and the uniqueness comes from a functional equation with a one-line solution.

That is worth holding onto, because it is the same shape of statement as the one about slopes. There, $e$ was the unique base whose curve is its own derivative. Here, $1/x$ is the unique curve whose area function turns products into sums. Both say the same thing — that this function is not a choice among many — and neither is visible from the other without the work above.

Which logarithm #

$L$ turns products into sums, so it is a logarithm — but every logarithm does that, and they differ by a constant factor. Which base is this one?

The base is the number whose logarithm is 1, and that is exactly what the hero figure locates: the point where the area first reaches 1, at $2.718282$. So $L$ is the logarithm to base $e$, which is why it is called the natural logarithm, and the naturalness is now visible rather than asserted. It is the logarithm that arises from a plain area with no constant stuck in front of it, and the strange-looking base is a consequence of that choice rather than a decision anybody made.

Every other base pays a conversion fee. The area interpretation of $\log_{10}$ is the area under $1/(x\ln 10)$, a curve with a peculiar constant in it chosen so that ten lands where $e$ used to be. The first essay in this ladder found the same fee in the language of slopes; here it is in the language of areas, and it is the same fee.

What had to be checked, and how #

A picture of a shaded region is a claim about a number, and a claim about a number invites the question of where the number came from. In every figure here the shaded area is computed by Simpson’s rule on twenty thousand strips under the drawn curve, and the built-in logarithm is used only afterwards, to check the answer.

That order matters more than it might seem. The whole essay is the assertion that an area and a logarithm are the same quantity, and a figure that drew the region and then labelled it $\log t$ would be illustrating the claim rather than testing it. Computing the area independently and requiring the two to agree to nine decimal places is a test that could fail, and a claim with no test it could fail is not the kind this collection makes.

The scaling argument gets the same treatment. The figure with two shaded regions computes both areas by quadrature and asserts that the second equals the area from 1 to the second factor — which is the geometric claim, checked arithmetically, with the geometry nowhere in the computation. The picture and the proof are independent of each other and are required to agree.

And back to slopes #

The two definitions must be connected, and the connection is the fundamental theorem: the rate at which an accumulating area grows is the height of the curve being accumulated.

Applied to $L$, whose integrand is $1/x$, this says

$$L'(t) = \frac1t.$$

Now invert. Let $E$ be the function undoing $L$, so that $L(E(x)) = x$. Differentiating both sides and using the chain rule gives $L'(E(x))\cdot E'(x) = 1$, and since $L'(E) = 1/E$,

$$E'(x) = E(x).$$

The inverse of the area function is its own derivative. That is the property the first essay defined $e$ by, recovered here as a theorem rather than assumed as a definition. The two starting points meet, and the meeting is the reflection that turns a slope into its reciprocal applied to one particular curve.

What this buys #

Three things, and the third is the reason it is standard.

It defines the exponential without circularity. The chain $b^x \to$ derivative $\to e$ has to explain what $b^x$ means for irrational $x$ before it can differentiate it, and the honest explanation goes through limits of rational powers and a continuity argument. The area chain has no such debt: $1/x$ is elementary, area is elementary, and $e^x$ is defined as the inverse of the area. Powers with arbitrary real exponents can then be defined as $a^b = E(b\,L(a))$, which is a definition rather than a limit.

It extends where the other definition does not. The area under $1/x$ makes sense for a complex path as well as a real interval, and taking it along different paths in the plane is where the multi-valuedness of the complex logarithm comes from — a subject with no counterpart at all in the story about slopes.

It gives the derivative of every power at once. With $a^b$ defined as $E(b\,L(a))$, the derivative of $x^n$ for arbitrary real $n$ comes out in one line and needs no separate treatment of whole numbers, fractions and irrationals. The route through repeated multiplication handles $x^3$ easily, $x^{1/2}$ with an argument about inverses, and $x^{\pi}$ not at all.

It connects to counting. The area under $1/x$ from 1 to $n$ is approximated by the sum $1 + 1/2 + \cdots + 1/n$, which is the harmonic series. The two differ by an amount that settles to a constant — the Euler–Mascheroni constant, 0.5772 — so the harmonic series grows like $\log n$, and every question about a sum of reciprocals becomes a question about this area.

The same area, read as a rate #

There is a third reading of the shaded region, and it is the one that explains why logarithms turn up in places with no multiplication in them.

The width of the strip needed to add a fixed amount of area at position $x$ is proportional to $x$, because the height there is $1/x$. So equal areas correspond to equal proportional increases: going from 1 to 2 and from 50 to 100 shade the same amount. The area is measuring not how far along the axis a point is but how many doublings it is from the start.

That is the whole of why a logarithmic scale is the right scale for a quantity that grows by ratios — populations, magnitudes, prices, orders of magnitude of anything. The scale is not a convenience for fitting large numbers onto a small axis; it is the scale on which equal distances mean equal factors, and the picture above is the reason there is such a scale at all.

It also explains a small puzzle about the harmonic series. That $1 + 1/2 + 1/3 + \cdots$ diverges is surprising because the terms go to nothing; that it diverges like a logarithm removes the surprise entirely. Adding terms from $n$ to $2n$ contributes about $\log 2$ every time, whatever $n$ is, so the sum climbs by a fixed amount over every doubling and doublings never run out. It diverges for the same reason the number line does.

What the picture cannot show #

The shaded region has an area of exactly 1 when its right edge is at $e$, and the figure prints $2.718282$ because that is as much as its arithmetic will carry. The picture cannot show that the edge does not land on a fraction — the irrationality of $e$ is proved from the series rather than seen — and it certainly cannot show that $e$ is transcendental, which is a statement about every polynomial with whole-number coefficients at once.

There is also a limit on what quadrature can settle. Twenty thousand strips give nine reliable decimal places and no more, so the figure can say that the edge is at 2.718282 and cannot say what the fiftieth digit is. That is a bound on the instrument rather than on the mathematics, but it is worth stating: a picture computed to nine places is evidence about nine places.

Nor can it show what happens to the left of 1, where the area is counted as negative and the picture would need a convention rather than a shading. The definition handles $t < 1$ perfectly well and the drawing does not, which is the usual gap between a definition and its illustration.

And it hides an assumption. That the area under $1/x$ exists — that the region has a well-defined area at all — is not obvious for a curve that runs off to infinity at the origin, and is only true because the right-hand edge stays away from zero. The area from 0 to 1 is infinite, and nothing in the shaded picture warns of the cliff just off its left edge.

The ladder from here #

Rungs above: the differential equation $y' = ky$, which picks the exponential out of every function there is. The exponential of a matrix, where the same series is summed with a square array in it. Where $e$ appears in counting, with no growth anywhere in sight. Euler’s identity as a statement about turning. The complex logarithm and the paths that make it multi-valued. And Stirling’s approximation, where $e$ turns up inside a factorial.

The habit #

The argument that carried the weight here is worth extracting from its setting.

Nothing was computed. A region was moved — stretched one way, squashed the other — and the two distortions were noticed to cancel. What made the argument work is that the curve $1/x$ is invariant under that pair of operations, so the moved region landed back under the same curve, and area was preserved because the two scalings were reciprocal.

That is the same move as reading a symmetry off a figure and concluding an equality, applied to a transformation that is not a symmetry of the plane but of one curve in it. Finding the transformations a particular object is unchanged by, and then asking what quantity they preserve, is a habit that will produce a great deal more than one logarithm.