Change of Base

Summary: §107–§108 of Chapter 6: any two systems of logarithms differ only by a multiplicative constant, so a single table can be converted to any other base by one multiplication. Equivalently (§108), the ratio of the logarithms of two given numbers is base-independent — it records the algebraic relationship between the numbers, not the choice of base.

Sources: chapter6 (§107–§108)

Last updated: 2026-04-26

---

The conversion rule (§107)

Let $n>0$ and let ${\log }_{a}n=p$, ${\log }_{b}n=q$ be its logarithms in two different systems. By definition $a^p=n=b^q$, hence $a^p=b^q$, so $a=b^{q/p}$. The ratio $p/q$ is therefore the same constant for every $n$ — call it $\lambda =p/q={\log }_{b}a$ (source: chapter6, §107).

Consequently: to convert a base-$a$ logarithm $p={\log }_{a}n$ to base $b$, multiply by $1/\lambda$:

${\log }_{b}n=\frac{p}{\lambda }=\frac{{\log }_{a}n}{{\log }_{a}b}.$

A single table of logarithms in any one system generates tables in every other system by one multiplication per entry.

Worked example: base 10 → base 2

From the standard tables, ${\log }_{10}2=0.3010300$ and ${\log }_{2}2=1$. So $p/q=0.3010300$, and the conversion factor is

$\frac{1}{{\log }_{10}2}=\frac{1}{0.3010300}=\mathrm{3.3219277.}$

Multiplying every common (base-10) logarithm by $3.3219277$ produces the corresponding base-2 logarithm (source: chapter6, §107). The “golden rule for logarithms,” in Euler’s phrase.

The base-free formulation (§108)

§108 reformulates the same fact without singling out a base. Take two numbers $M,N$. In base $a$, write $M=N^{m/n}$ (so ${\log }_{a}M/{\log }_{a}N=m/n$). In base $b$, write $M=N^{\mu /\nu }$ with the same $M,N$. Then

$N^{m/n}=M=N^{\mu /\nu }⟹\frac{m}{n}=\frac{\mu }{\nu },$

so

$\frac{\log M}{\log N}=\frac{m}{n},\text{independent of base.}$

In words: the ratio of the logarithms of two numbers is intrinsic — it is the rational (or transcendental) exponent expressing one number as a power of the other.

For powers of a common base, $y^m/y^n$ has $\log y^m/\log y^n=m/n$ — so log ratios of powers of a fixed number reduce to ratios of exponents.

Consequences

• There is essentially one logarithm function, parameterized by base: every choice differs from every other by a multiplicative scalar.
• Equations of the form $a^x=b$ have the base-independent solution $x=\log b/\log a$ (source: chapter6, §111) — the ratio is the same in any system, so one can use the most convenient table.
• Computational interchange is trivial: a base-10 table built once (e.g. by the geometric-mean method) suffices for any other base via one multiplication.

§109 — Tables built from primes

A direct corollary of the product rule and §107: only the logarithms of primes need be computed by the slow geometric-mean method. Logarithms of composites follow by addition (source: chapter6, §109):

$\log 15=\log 3+\log 5,\log 45=2\log 3+\log 5,\log 4=2\log 2,\dots$

And $\log 2$ itself can be deduced from $\log 5$ by $\log 2=\log 10-\log 5=1-0.6989700=0.3010300$ (in base 10), so a single root extraction (for $\log 5$) supplies both prime logs needed to tabulate all numbers whose only prime factors are 2 and 5.

Related pages

• logarithm
• geometric-mean-method-for-logarithms
• common-logarithm
• characteristic-and-mantissa
• chapter-6-on-exponentials-and-logarithms