Calculus I · The Fundamental Theorem of Calculus

The capstone: defining the logarithm, and settling every debt

We end by paying our debts. All through this book we borrowed: in Chapter 1 we asserted the number $e$ exists; in Chapter 5 we used $\ln' x = \tfrac1x$ and the laws of logarithms; each time we flagged it as a loan against a later, rigorous account. This is that account. And the instrument that settles everything is the one antiderivative the power rule could never produce — the missing $n=-1$ case, $\int \tfrac1x\,dx$.

So we turn it around and make it a definition:

$$\boxed{\;\ln x \;:=\; \int_1^x \frac{1}{t}\,dt \quad (x > 0).\;}$$

We define the natural logarithm to be the area under $\tfrac1t$ from $1$ to $x$. From this one line, with nothing assumed, every property of $\ln$ we ever used falls out as a theorem:

• $\ln 1 = 0$: it’s $\int_1^1 \tfrac1t\,dt$, an integral over no width. ✓ (The loan that $\ln 1 = 0$ — repaid by definition.)
• $\ln' x = \tfrac1x$: immediate from the Fundamental Theorem, Part 1 — the derivative of $\int_1^x \tfrac1t\,dt$ is the integrand $\tfrac1x$. The fact we assumed in Chapter 5 is now a one-line consequence. ✓
• $\ln$ is increasing for $x>0$: its derivative $\tfrac1x$ is positive. And it grows without bound as $x\to\infty$ (the area under $\tfrac1t$ never stops accumulating). ✓
• The product law $\ln(ab) = \ln a + \ln b$: split the defining integral and substitute. We have $\ln(ab) = \int_1^{ab}\tfrac1t dt = \int_1^a \tfrac1t dt + \int_a^{ab}\tfrac1t dt$. In the second piece let $u = t/a$ (so $dt = a\,du$, and $t:a\to ab$ becomes $u:1\to b$): it turns into $\int_1^b \tfrac{1}{u}\,du = \ln b$. Hence $\ln(ab) = \ln a + \ln b$ — the signature property of logarithms, emerging from integral additivity and a substitution. ✓
• The number $e$ exists: define $e$ as the unique number with $\ln e = 1$ — i.e. the right edge at which the area under $\tfrac1t$ first equals $1$. It exists because $\ln$ is continuous with $\ln 1 = 0 < 1$ and grows past $1$, so by the Intermediate Value Theorem (Chapter 2) it hits $1$ somewhere. This is the very loan from Chapter 1 — that $e$ exists at all — now discharged completely. ✓
• $e^x$ and its derivative: the exponential is simply the inverse function of $\ln$, and $\tfrac{d}{dx}e^x = e^x$ follows from the inverse-function rule (Chapter 4). ✓

Every promissory note the book ever wrote is now paid. The whole edifice — $e$, $\ln$, $e^x$, their derivatives and laws — which we used freely and honestly flagged as borrowed, rests in the end on a single integral and the Fundamental Theorem. The account balances. (Figure 11.2: the curve $y = \tfrac1t$ with the area from $1$ to $x$ shaded and labeled $\ln x$; a second marker at $t=e$ where the accumulated area first reaches exactly $1$.)

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