Exponential Function

Summary: Euler’s exponential is the function $y=a^z$ with constant base $a$ and variable exponent $z$ (§96–§101). Defined first on integers, then on rationals by $a^{p/q}=\sqrt[q]{a^p}$ (taking the primary positive value), then extended by interpolation to irrational $z$. Restricting to $a>1$ gives the canonical case: a single-valued, strictly increasing function from $\mathbb{R}$ onto $(0,\infty )$.

Sources: chapter6 (§96–§101)

Last updated: 2026-04-26

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Why exponentials are not algebraic

A function built only from arithmetic operations and root extraction with constant exponents is algebraic (see classification-of-functions). The moment the exponent becomes a variable — as in $a^z$, $y^z$, $a^{a^z}$, $a^{y^z}$, $y^{x^z}$, $x^{y^z}$ — the function leaves the algebraic class (source: chapter6, §96). Euler treats $a^z$ as the canonical representative because the analysis of one variant settles the others.

Definition by extension

Let $a>0$. The exponential $a^z$ is defined in stages (source: chapter6, §97):

• Positive integer $z$: $a^z=a\cdot a\cdots a$ ($z$ factors).
• $z=0$: $a^0=1$.
• Negative integer $z=-n$: $a^{-n}=1/a^n$.
• Rational $z=p/q$: $a^{p/q}=\sqrt[q]{a^p}$. The $q$-th root is generally multi-valued, but Euler restricts to the primary positive real value so that $a^z$ is single-valued. So $a^{5/2}$ lies between $a^2$ and $a^3$, and equals $a^2\sqrt{a}$ (not its negative).
• Irrational $z$: by interpolation. $a^{\sqrt{7}}$ lies between $a^2$ and $a^3$ and is determined as the common value of all rational approximations from above and below. Euler does not prove this is well-defined; he treats it as evident from the monotonic interpolation.

Restriction to real exponents is explicit: complex $z$ is deferred to later chapters.

Case analysis on the base

The qualitative behavior of $a^z$ depends on $a$ (source: chapter6, §98–§99):

Range of $a$	Behavior of $a^z$
$a=1$	constant $1$
$a>1$	strictly increasing; $a^z\to \infty$ as $z\to \infty$, $a^z\to 0$ as $z\to -\infty$
$0<a<1$	strictly decreasing; reduces to $a>1$ via $1/a$, since $a^z=(1/a{)}^{-z}$
$a=0$	$0^z=0$ for $z>0$; $0^0=1$; $0^{-n}=1/0^n$ “is infinite” — discontinuous at $z=0$
$a<0$	integer $z$: alternating sign; rational $z$: real or pure-imaginary depending on parity ($(-2{)}^{1/2}=\sqrt{-2}$, $(-2{)}^{1/3}=-\sqrt[3]{2}$); irrational $z$: unpredictable

§100 distills the conclusion: take $a>1$ as the canonical case. The case $0<a<1$ then reduces via $1/a$, and $a\le 0$ is set aside as pathological.

Properties (§101)

For $y=a^z$ and $v=a^x$:

$y^n=a^{nz},y^{1/n}=a^{z/n},\frac{1}{y}=a^{-z},$

$vy=a^{x+z},\frac{v}{y}=a^{x-z}.$

These multiplicative-to-additive identities are what make logarithms (the inverse function — see logarithm) so useful.

For the example $a=10$: $10^1=10$, $10^2=100$, $10^{-1}=0.1$, $10^{1/2}=\sqrt{10}\approx 3.162277$, etc.

Shape of the canonical case

When $a>1$:

• Domain: all real $z$.
• Range: $(0,+\infty )$.
• Strictly increasing, continuous, single-valued.
• Crosses $1$ at $z=0$.
• Asymptote $y=0$ as $z\to -\infty$; unbounded as $z\to \infty$.

Every positive $y$ has a unique real $z$ with $a^z=y$ — this is what makes the inverse $\log y$ well-defined as a real-valued function on $(0,\infty )$.

Related pages

• logarithm
• classification-of-functions
• function
• chapter-6-on-exponentials-and-logarithms