Logarithm

Summary: For base $a>1$, the logarithm of $y>0$ is the unique real $z$ such that $a^z=y$, written $z=\log y$ (§102). Logarithms convert multiplication into addition (§104) and turn the algebra of exponentials inside out. They are real-valued only for positive arguments; for “generic” arguments they are transcendental — see transcendence-of-logarithms.

Sources: chapter6 (§102–§104)

Last updated: 2026-04-26

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Definition

Fix a base $a>1$. For each $y>0$, the exponential-function $a^z$ takes the value $y$ exactly once, so the equation $a^z=y$ has a unique real solution $z$. This $z$ is called the logarithm of $y$ to the base $a$, and is denoted

$z=\log y\text{when}{a}^z=y$

(source: chapter6, §102). The base must be specified for the symbol $\log$ to be unambiguous; “infinitely many systems of logarithms” exist, one for each base (see change-of-base).

Euler restricts to bases $a>1$ throughout the chapter — the only setting in which $\log$ is real-valued on $(0,\infty )$ and increasing.

When the logarithm is real

• $y>0$: $\log y$ is a real number.
• $y=0$: no real $z$ satisfies $a^z=0$ (the equation has $z=-\infty$ as a limit, not a value).
• $y<0$: no real $z$ satisfies $a^z=y$ when $a>1$; “the logarithm is complex” (source: chapter6, §103).

First values (§103)

$\log 1=0,\log a=1,\log a^2=2,\log a^n=n,$

and

$\log \frac{1}{a}=-1,\log \frac{1}{a^2}=-2,\log \frac{1}{a^n}=-n.$

So $\log y>0$ for $y>1$ and $\log y<0$ for $0<y<1$, in any system. The base $a$ itself is identified after the fact as the unique number whose logarithm is 1.

Algebraic rules (§104)

The properties of the exponential-function translate into four rules:

Exponential identity	Logarithmic identity
$y^n=a^{nz}$	$\log y^n=n\log y$
$\sqrt[n]{y}=a^{z/n}$	$\log \sqrt[n]{y}=\frac{1}{n}\log y$
$vy=a^{x+z}$	$\log (vy)=\log v+\log y$
$v/y=a^{x-z}$	$\log (v/y)=\log v-\log y$

These four are the entire algebra of logarithms (source: chapter6, §104). Their power: a complicated arithmetic expression in $v,y,n,\dots$ becomes a linear combination of the logarithms $\log v,\log y,\dots$ — the trick that makes logarithm tables a universal calculation tool (cf. §110).

Consequences

• Logarithms of powers and roots of a known number $y$ follow from $\log y$ alone.
• Logarithms of products and quotients of known numbers reduce to sums and differences.
• A table of $\log p$ for each prime $p$ therefore generates the logarithm of every positive rational by additions, subtractions, and rational multiples (see change-of-base and §109).

Why “transcendental” (§105)

Most logarithms are not rational, not even algebraic — see transcendence-of-logarithms. This forces them to be computed approximately, and motivates the geometric-mean algorithm of §106 and the table-driven calculations of §110–§111.

Related pages

• exponential-function
• transcendence-of-logarithms
• geometric-mean-method-for-logarithms
• change-of-base
• common-logarithm
• characteristic-and-mantissa
• chapter-6-on-exponentials-and-logarithms