Chapter 6: On Exponentials and Logarithms

Summary: Euler’s first transcendental chapter. He develops the exponential-function $y=a^z$ informally — extending from integer to fractional to irrational exponents — and then introduces its inverse, the logarithm. Most logarithms are transcendental (§105); they are computed in practice by iterated geometric means (§106), tabulated for primes (§109), and converted between bases by a single multiplicative constant (§107–§108). The chapter closes with the working machinery of common (base-10) logarithms: characteristic-and-mantissa.

Sources: chapter6

Last updated: 2026-04-26

---

Overview

Up to Chapter 5 every function was algebraic — built from the four arithmetic operations and root extraction (see classification-of-functions). Chapter 6 takes the first step into transcendental territory. Euler concedes that a fully rigorous theory needs integral calculus, but two species of transcendental function — exponentials and their inverse logarithms — can already be developed by elementary means, and they “open the door to further investigations” (source: chapter6, §96).

The chapter has three movements:

1. §96–§101 — the exponential. Define $y=a^z$ for variable exponent. Walk through integer, fractional, irrational $z$ to make plausible that the function is single-valued and continuous. Case-split on $a$ to settle on $a>1$ as the canonical setting.
2. §102–§109 — the logarithm. Invert: $\log y$ is the exponent that produces $y$. Derive the algebraic rules. Show (§105) that logarithms of “generic” numbers are transcendental. Sketch the computational machinery: compute by iterated geometric means (§106), change base by a single multiplier (§107–§108), and reduce all tabulation to the primes (§109).
3. §110–§113 — applications and decimal logarithms. Use logarithms to compute complicated expressions and solve $a^x=b$; work out four word problems on population growth and compound interest. Then define common logarithms (base 10) and their split into characteristic and mantissa.

See also: exponential-function, logarithm, transcendence-of-logarithms, geometric-mean-method-for-logarithms, change-of-base, common-logarithm, characteristic-and-mantissa.

Structure of the chapter

§96 — Why exponentials are not algebraic

A power with variable exponent — $a^z$, $y^z$, $a^{a^z}$, $a^{y^z}$, $y^{x^z}$, $x^{y^z}$ — is not an algebraic function, since algebraic functions require the exponents to be constants (source: chapter6, §96). Euler absorbs all the variant forms into a single representative $a^z$: the analysis of one settles the others.

§97 — Defining $a^z$ on $\mathbb{Q}$

For positive integer $z$, $a^z$ has the obvious meaning. For $z=0$, set $a^0=1$. For negative integer $z$, $a^{-n}=1/a^n$. For rational $z=p/q$, $a^{p/q}=\sqrt[q]{a^p}$, which is in general multivalued, but Euler restricts to the primary positive real value so that $a^z$ is single-valued (source: chapter6, §97). This places, e.g., $a^{5/2}$ between $a^2$ and $a^3$. Irrational $z$ is handled by interpolation: $a^{\sqrt{7}}$ lies between $a^2$ and $a^3$. See exponential-function.

§98–§99 — Case analysis on the base

The behavior of $a^z$ depends on $a$ (source: chapter6, §98):

• $a=1$: constant 1.
• $a>1$: $a^z$ strictly increases; $a^z\to \infty$ as $z\to \infty$ and $a^z\to 0$ as $z\to -\infty$.
• $0<a<1$: write $1/a=b>1$ to get $a^z=b^{-z}$ — the $a>1$ case, reflected.
• $a=0$: discontinuity at $z=0$. For $z>0$, $a^z=0$; for $z=0$, $a^0=1$; for $z<0$, $a^{-n}=1/0^n$ “is infinite” (source: chapter6, §99).
• $a<0$: integer exponents alternate sign; rational exponents may produce real or pure-imaginary values ($(-2{)}^{1/2}=\sqrt{-2}$ vs. $(-2{)}^{1/3}=-\sqrt[3]{2}$); irrational exponents are unpredictable.

§100 distills the conclusion: take $a>1$; the case $0<a<1$ then follows by reflection, and the others are pathological.

§101 — Algebraic rules and the example $a=10$

With $y=a^z$: $y^n=a^{nz}$, $y^{1/n}=a^{z/n}$, $1/y=a^{-z}$; if $v=a^x$ then $vy=a^{x+z}$ and $v/y=a^{x-z}$ (source: chapter6, §101). Worked example for $a=10$: $10^1=10$, $10^2=100$, $10^{-1}=0.1$, $10^{1/2}=\sqrt{10}\approx 3.162277$, etc.

§102 — Logarithm as inverse exponent

For each $y>0$ there is a unique real $z$ with $a^z=y$; this $z$ is called the logarithm of $y$ to the base $a$, written $z=\log y$. The base $a$ must be specified for the symbol to be unambiguous, and Euler assumes $a>1$ throughout (source: chapter6, §102). See logarithm.

§103 — First values of $\log$

$\log 1=0$ regardless of base; $\log a=1$, $\log a^2=2$, $\log a^3=3$, etc.; $\log (1/a^n)=-n$ (source: chapter6, §103). Numbers $>1$ have positive logs; numbers in $(0,1)$ have negative logs; the “logarithm” of a negative number is complex.

§104 — The algebraic rules of logarithms

From $\log y^n=n\log y$, one gets $\log \sqrt{y}=\frac{1}{2}\log y$, $\log (1/\sqrt{y})=-\frac{1}{2}\log y$, etc. From $vy=a^{x+z}$ one gets $\log (vy)=\log v+\log y$ and $\log (v/y)=\log v-\log y$ (source: chapter6, §104). These four rules — log of a power, log of a root, log of a product, log of a quotient — are the entire algebra of logarithms.

§105 — Logarithms are transcendental

Suppose $\log b$ is rational, say $\log b=m/n$. Then $a^{m/n}=b$, i.e. $a^m=b^n$; if both $a$ and $b$ are rational this forces $b$ to be a rational power of $a$. Suppose instead $\log b$ is irrational, say $\log b=\sqrt{n}$. Then $a^{\sqrt{n}}=b$ — impossible if $a,b$ are rational (source: chapter6, §105). Conclusion: unless $b$ is exactly a power of $a$, $\log b$ is neither rational nor irrational algebraic — Euler labels such quantities transcendental, and so logarithms in general are transcendental. See transcendence-of-logarithms.

§106 — Computing logarithms by geometric means

A transcendental logarithm can be approximated to arbitrary decimal precision by an algorithm using only square roots. The principle: if $\log y=z$ and $\log v=x$, then $\log \sqrt{vy}=(x+z)/2$ — the log of the geometric mean is the arithmetic mean of the logs (source: chapter6, §106).

To find ${\log }_{10}5$: 5 lies between $A=1$ ($\log A=0$) and $B=10$ ($\log B=1$). Set $C=\sqrt{AB}=3.162277$, $\log C=0.5$. Now 5 lies between $C$ and $B$; take $D=\sqrt{BC}$, etc. The bounds halve at each step. After ~26 iterations the geometric mean stabilizes to $5.000000$ with $\log =0.6989700$.

This is how Briggs and Vlacq computed the original tables of common logarithms (source: chapter6, §106 example). See geometric-mean-method-for-logarithms.

§107–§108 — Change of base

Different bases give different logarithm systems, but they are all proportional. If ${\log }_{a}n=p$ and ${\log }_{b}n=q$, then $a^p=n=b^q$, so $a=b^{q/p}$, and the ratio $p/q$ is the same for every $n$ (source: chapter6, §107). Thus a single multiplier converts one full table into another: e.g. base-10 to base-2 by multiplying every common log by $1/{\log }_{10}2=1/0.3010300=3.3219277$.

§108 strengthens this with a base-free formulation: for any two numbers $M,N$ in the same system, the ratio of their logarithms is base-independent. From $M=N^{m/n}$ in base $a$ and $M=N^{\mu /\nu }$ in base $b$, both ratios equal — they record the algebraic relationship between $M$ and $N$, not the choice of base. See change-of-base.

§109 — Tables built from primes

The product rule reduces tabulation to prime logs: $\log 15=\log 3+\log 5$, $\log 4=2\log 2$, etc. Once $\log 2=0.3010300$ and $\log 5=0.6989700$ are computed, every number whose only prime factors are 2 and 5 (i.e. every terminating decimal multiplied by a power of 10) follows by addition (source: chapter6, §109). Note also that $\log 2=\log 10-\log 5=1-0.6989700$, so a single root extraction (for $\log 5$) suffices for both.

§110–§111 — Applications

§110 uses logs to evaluate complicated algebraic expressions: e.g. $\log \left(\frac{c^{2}d\sqrt{e}}{f(gh{)}^{1/3}}\right)=2\log c+\log d+\frac{1}{2}\log e-\log f-\frac{1}{3}\log g-\frac{1}{3}\log h$. Numerical roots are found this way without explicit root extraction.

§111 turns to the most important application: solving equations with the unknown in the exponent. From $a^x=b$, $x=\log b/\log a$ — the choice of base is irrelevant since the ratio is invariant (§107–§108). Worked examples:

• Population growth. If a population grows by $1/30$ per year, after 100 years it is multiplied by $(31/30{)}^{100}$. Working in common logs, $100\log (31/30)=1.4240439$, so the population grows by a factor of $\approx 26.5$ (Example II, §110).
• Bible-flood example. Starting from 6 people after the Flood and reaching 1,000,000 after 200 years requires an annual growth rate of about $1/16$; “the same rate over 400 years gives 166 billion — but the earth could not sustain it” (Example III, §110).
• Doubling per century. Annual rate is then $1/144$ (Example IV, §110).
• Time to a tenfold population. At rate $1/100$ per year, $x=\log 10/(\log 101-\log 100)\approx 231$ years (Example I, §111).
• Compound debt. A loan of 400,000 florins at 5% per annum, repaid 25,000 per year, is settled in just under 33 years — the creditor in fact owes the debtor 318.8 florins after 33 years (Example II, §111).

§112–§113 — Common logarithms: characteristic and mantissa

Base 10 is special because our arithmetic is decimal. A common logarithm splits into:

• Characteristic: the integer part. Equal to (number of digits in the integer part) $-1$. So $\log 78509$ has characteristic 4; reading the characteristic of any $\log$, one knows the number’s digit count (source: chapter6, §112).
• Mantissa: the decimal fractional part. Encodes the digit string of the number, independent of decimal placement (source: chapter6, §113).

Two numbers whose logs share a mantissa differ only by a power of 10 — same digits, decimal point shifted. Negative characteristics are conventionally shifted upward by 10 (writing 9, 8, 7, … for $-1,-2,-3$ and noting “diminished by 10”). See characteristic-and-mantissa and common-logarithm.

The closing example: in the progression $2,4,16,256,\dots$ where each term is the square of the previous, the 25th term is $2^{2^{24}}=2^{16777216}$, whose common logarithm is $16777216\cdot \log 2=5050445.25973367$. The characteristic 5050445 says the number has 5,050,446 digits, and the mantissa locates its leading digits — Euler reports the eleven leading digits as $18185852986$.

Notable points

• First definition of “transcendental” applied to a number, not a function. §105 names logarithms transcendental — extending the algebraic/transcendental distinction from classification-of-functions from expressions to quantities. The argument is a clean dichotomy: either rational (force on $b$) or impossible.
• The geometric-mean algorithm (§106) is striking. It computes a transcendental quantity from purely algebraic operations (square roots and bisection of brackets). This anticipates the quadrature-style computations of subsequent chapters and is conceptually parallel to how $e$ and $\pi$ will be approached later.
• Change of base (§107–§108) reduces “infinitely many systems of logarithms” to one. Logarithms are intrinsically a single one-parameter family — the only base-dependent thing is a multiplicative constant. The base-free §108 formulation ($\log M:\log N$ is base-invariant) is the proportionality principle that ties all the systems together.
• The Bible-flood example (§110, Example III) is unusual. Most of Euler’s worked examples are abstract; this one is socio-theological. He uses it to defend the plausibility of biblical population numbers — a 1/16 annual growth rate is enough to take the post-Flood population from 6 to 1,000,000 in 200 years.
• §113’s last example is a flex. Computing the digit count of $2^{16777216}$ shows logarithms doing work no other tool can do at the time — the number itself is uncomputable in any direct sense.
• No mention of $e$ yet. Euler is careful: the special role of base $e$ requires the limit ${\lim }_n(1+1/n{)}^n$, which belongs to Chapter 7. Here every base is on equal footing.

Why this chapter matters

Chapter 6 introduces the first transcendental functions and the first numerical constants outside the algebraic realm. The algebraic theory of Chapters 1–5 was self-contained but bounded: rational functions, polynomial roots, partial fractions, homogeneous reductions. Now Euler crosses into a class of functions whose values cannot in general be expressed by finite algebraic formulas — and supplies, through §106 and §107–§109, exactly the computational scaffolding (geometric means, change of base, prime tables, characteristic/mantissa) that makes such functions usable in practice.

The chapter is also a setup. Chapter 7 will return to $a^z$, expand it as a power series via the limit ${\lim }_n(1+z/n{)}^n$, identify the privileged base $e$ for which the series is simplest, and so launch the analytic theory of exponentials and logarithms that the rest of Book I depends on.

Related pages

• exponential-function
• logarithm
• transcendence-of-logarithms
• geometric-mean-method-for-logarithms
• change-of-base
• common-logarithm
• characteristic-and-mantissa
• classification-of-functions
• chapter-4-on-the-development-of-functions-in-infinite-series