Logarithms

Summary: Logarithms are the exponents needed to produce a given number from a fixed base, and they transform multiplication, division, powers, and roots into simpler additive operations. (source: chapter-1.1.20, source: chapter-1.1.21, source: chapter-1.1.22, source: chapter-1.1.23)

Sources: chapter-1.1.20, chapter-1.1.21, chapter-1.1.22, chapter-1.1.23

Last updated: 2026-04-24

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Definition

For a fixed base $a$, the logarithm of $c$ is the exponent $b$ such that: $a^b=c.$ (source: chapter-1.1.21)

Euler introduces logarithms as the final inverse operation in the chain that begins with addition and multiplication and continues through powers and roots. (source: chapter-1.1.20)

Main Laws

The governing identities are: $\log (cd)=\log c+\log d,\log \left(\frac{c}{d}\right)=\log c-\log d,\log (c^n)=n\log c.$ (source: chapter-1.1.21)

These laws make roots equally manageable: $\log \left(\sqrt[n]{c}\right)=\frac{1}{n}\log c.$ (source: chapter-1.1.21)

Common Base-10 Tables

Euler then specializes to base $10$, so: $10^{\log c}=c.$ (source: chapter-1.1.22)

In this system: $\log 1=0,\log 10=1,\log 100=2.$ (source: chapter-1.1.22)

Practical Use

Tabular logarithms are written in decimal form and used to replace multiplication by addition, division by subtraction, powers by multiplying the logarithm, and roots by dividing it. (source: chapter-1.1.23)

Euler also remarks that logarithms of negative numbers are impossible and belong to imaginary quantities. (source: chapter-1.1.21)

Related pages

• ch1.1.20-methods-of-calculation
• ch1.1.21-logarithms-in-general
• decimal-fractions
• powers-and-exponents
• roots
• imaginary-numbers