ch1.1.21-logarithms-in-general

Chapter XXI – Of Logarithms in general

Summary: Defines logarithms as exponents, derives the basic laws $\log (cd)=\log c+\log d$ and $\log (c/d)=\log c-\log d$, and explains why logarithms simplify multiplication, division, powers, and roots. (source: chapter-1.1.21)

Sources: chapter-1.1.21

Last updated: 2026-04-24

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§220–224: Definition

Starting from $a^b=c$, Euler defines $b$ as the logarithm of $c$ for the fixed base $a$: $b=\log c\text{when}{a}^b=c.$ (source: chapter-1.1.21)

He notes that: $\log a=1,\log a^2=2,\log a^3=3,\log 1=0.$ (source: chapter-1.1.21)

Negative exponents correspond to logarithms of reciprocals, and fractional exponents correspond to logarithms of roots. (source: chapter-1.1.21)

§225–226: Fundamental Laws

From the laws of exponents, Euler derives: $\log (cd)=\log c+\log d,\log \left(\frac{c}{d}\right)=\log c-\log d.$ (source: chapter-1.1.21)

These are the principal identities on which logarithmic calculation rests. (source: chapter-1.1.21)

§227–229: Use in Computation

Because products and quotients turn into sums and differences of logarithms, logarithms shorten difficult numerical work. (source: chapter-1.1.21)

Euler also derives: $\log (c^n)=n\log c,\log \left(\sqrt[n]{c}\right)=\frac{1}{n}\log c.$ (source: chapter-1.1.21)

Thus powers can be found by multiplication of logarithms and roots by division of logarithms. (source: chapter-1.1.21)

§230–231: Sign of the Logarithm and Choice of Base

Positive logarithms correspond to numbers greater than $1$, while negative logarithms correspond to positive fractions less than $1$. (source: chapter-1.1.21)

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

He then notes that the base may be any positive number greater than $1$, though the next chapter specializes to base $10$. (source: chapter-1.1.21)

Related pages

• logarithms
• powers-and-exponents
• imaginary-numbers
• decimal-fractions