ch1.1.22-logarithmic-tables

Chapter XXII – Of the Logarithmic Tables now in use

Summary: Specializes logarithms to base $10$, explains how tabular logarithms of many numbers can be derived from a few known ones, and emphasizes prime factorization as the key to constructing tables. (source: chapter-1.1.22)

Sources: chapter-1.1.22

Last updated: 2026-04-24

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§232–233: Base-10 Logarithms

Euler now fixes the base at $10$, so logarithms satisfy: $10^{L(c)}=c.$ (source: chapter-1.1.22)

This gives immediately: $\log 1=0,\log 10=1,\log 100=2,\log 1000=3,$ and likewise: $\log \frac{1}{10}=-1,\log \frac{1}{100}=-2,\log \frac{1}{1000}=-3.$ (source: chapter-1.1.22)

§234–236: Locating $\log 2$

Euler shows qualitatively how one can bracket the logarithm of $2$ between fractions by comparing powers of $10$ with powers of $2$. He concludes that $\log 2$ has a determinate value between tested bounds even if its exact decimal form has not yet been computed. (source: chapter-1.1.22)

§237–240: Deducing Many Logarithms from a Few

If $\log 2=x$, then Euler derives families such as: $\log 20=x+1,\log 4=2x,\log 5=1-x.$ (source: chapter-1.1.22)

If $\log 3=y$, he obtains similarly: $\log 9=2y,\log 6=x+y,\log 15=y+1-x.$ (source: chapter-1.1.22)

§241: Prime Numbers as the Basis of the Tables

Because every integer factors into primes, Euler concludes that logarithms of all numbers can be built from logarithms of prime numbers by addition. (source: chapter-1.1.22)

He illustrates this with numbers such as $210=2\cdot 3\cdot 5\cdot 7$ and $360=2^3\cdot 3^2\cdot 5$. (source: chapter-1.1.22)

Related pages

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