Natural Logarithm

Summary: §123–§125 of Chapter 7. The natural (or hyperbolic) logarithm is $\log$ in the base $a=e$ of eulers-number — the unique base for which $\log (1+\omega )=\omega$ for infinitely small $\omega$ (equivalently, the constant $k=1$ in §116). Euler tabulates ${\log }_{e}n$ for $n=1,\dots ,10$ to twenty decimal places using the fast-converging series $\log \frac{1+x}{1-x}=2(x+x^3/3+x^5/5+\cdots )$, then shows that for any other base $a$, $k={\log }_{e}a$ is the conversion factor — so a single table of natural logs supplies every other system by one multiplication, recovering §107–§108 from a different angle.

Sources: chapter7 (§123–§125)

Last updated: 2026-04-26

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Defining property (§123)

Natural logarithms have the property that the logarithm of $1+\omega$ is equal to $\omega$, where $\omega$ is an infinitely small quantity.

(source: chapter7, §123). This is exactly the condition $k=1$ from §114 — and it picks out the base $a=e$ uniquely. Equivalently:

• $a^{\omega }=1+\omega$ for infinitely small $\omega$ — the exponential is “tangent to $1+z$ at $z=0$” in modern language;
• the exponential series is $e^z=1+z+z^2/2+z^3/6+\cdots$ with no extra factor;
• the log series is $\log (1+x)=x-x^2/2+x^3/3-\cdots$ with no extra factor.

The three master series in base $e$

With $k=1$ the series of §116 and §119, §121 take their cleanest forms (source: chapter7, §123):

$e^z=1+\frac{z}{1}+\frac{z^2}{1\cdot 2}+\frac{z^3}{1\cdot 2\cdot 3}+\frac{z^4}{1\cdot 2\cdot 3\cdot 4}+\cdots$

$\log (1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\frac{x^5}{5}-\frac{x^6}{6}+\cdots$

$\log \frac{1+x}{1-x}=\frac{2x}{1}+\frac{2x^3}{3}+\frac{2x^5}{5}+\frac{2x^7}{7}+\frac{2x^9}{9}+\cdots$

The third series is “strongly convergent if we substitute an extremely small fraction for $x$” (source: chapter7, §123) and is the workhorse for the table below.

The integer table (§123)

Using $x=1/5,1/7,1/9$ in $\log \frac{1+x}{1-x}$:

$\log \frac{6}{4}=\log \frac{3}{2}=\frac{2}{1\cdot 5}+\frac{2}{3\cdot 5^3}+\frac{2}{5\cdot 5^5}+\frac{2}{7\cdot 5^7}+\cdots$

$\log \frac{4}{3}=\frac{2}{1\cdot 7}+\frac{2}{3\cdot 7^3}+\frac{2}{5\cdot 7^5}+\frac{2}{7\cdot 7^7}+\cdots$

$\log \frac{5}{4}=\frac{2}{1\cdot 9}+\frac{2}{3\cdot 9^3}+\frac{2}{5\cdot 9^5}+\frac{2}{7\cdot 9^7}+\cdots$

Combining via the algebraic rules:

$\log \frac{3}{2}+\log \frac{4}{3}=\log 2,\log \frac{3}{2}+\log 2=\log 3,2\log 2=\log 4,\log \frac{5}{4}+\log 4=\log 5,$

$\log 2+\log 3=\log 6,3\log 2=\log 8,2\log 3=\log 9,\log 2+\log 5=\log 10.$

For $\log 7$, $x=1/99$ gives $\log (100/98)=\log (50/49)=0.0202027073175194484078230$. Subtracting from $\log 50=2\log 5+\log 2=3.9120230054281460586187508$ gives $\log 49$, and $\log 7=\frac{1}{2}\log 49$.

The resulting table (source: chapter7, §123 example), to twenty digits:

$n$	$\log n=\ln n$
1	$0.0000000000000000000000000$
2	$0.6931471805599453094172321$
3	$1.0986122886681096913952452$
4	$1.3862943611198906188344642$
5	$1.6094379124341003746007593$
6	$1.7917594692280550008124773$
7	$1.9459101490553133051054639$
8	$2.0794415416798359282516964$
9	$2.1972245773362193827904905$
10	$2.3025850929940456840179914$

These are the natural logarithms — the modern $\ln 1,\ln 2,\dots ,\ln 10$.

$k={\log }_{e}a$ as the change-of-base factor (§124)

Suppose the natural log of $1+x$ is $y$. By §123,

$y=x-\frac{x^2}{2}+\frac{x^3}{3}-\cdots$

If $v$ is the logarithm of the same $1+x$ in base $a$, then by §119,

$v=\frac{1}{k}\left(x-\frac{x^2}{2}+\frac{x^3}{3}-\cdots \right)=\frac{y}{k}.$

So $k=y/v$. Setting $1+x=a$ makes $v={\log }_{a}a=1$ and $y={\log }_{e}a$:

$\boxed{k={\log }_{e}a.}$

This is “the most convenient method of calculating the value of $k$ corresponding to the base $a$” (source: chapter7, §124). For $a=10$:

$k={\log }_{e}10=2.3025850929940456840179914,$

the same value computed in §114 from the table and in §121 from the fast-converging series.

The reciprocal,

$\frac{1}{k}={\log }_{10}e=0.4342944819032518276511289,$

multiplies a natural log to produce a common log, and is the famous “modulus” of common logarithms.

$a^y=e^{y\log a}$ (§125)

Substituting $a^y=e^z$ with $z=y\log a$ (since $\log e=1$ implies $\log a^y=y\log a$ in natural logs) into $e^z=\sum z^n/n!$:

$a^y=1+\frac{y\log a}{1}+\frac{y^2(\log a{)}^2}{1\cdot 2}+\frac{y^3(\log a{)}^3}{1\cdot 2\cdot 3}+\cdots$

— the §117 formula in disguise, with $k=1$ absorbed and $\log ={\log }_e$ everywhere (source: chapter7, §125).

The two infinite-power forms recap:

$e^z={\left(1+\frac{z}{j}\right)}^j,a^y={\left(1+\frac{y\log a}{j}\right)}^j,\log (1+x)=j((1+x{)}^{1/j}-1),$

with $j$ infinitely large — see infinitesimal-and-infinite-numbers.

Why “natural” is the right word

Two clean facts pick out base $e$:

• Tangency at the identity: $\log (1+\omega )=\omega$ for infinitely small $\omega$ — the natural log is the only one whose graph has slope 1 at $1$. (Modern: $(d/dx){\log }_{a}x{|}_{x=1}=1$ iff $a=e$.)
• Inverse with itself: $e^z$ and ${\log }_e$ are the unique base-pair for which both the exponential and logarithm series have unit leading coefficient.

Every other base introduces the multiplicative constant $k$, $1/k$, or $\log a$ somewhere. Natural logs are “natural” in the sense of being unweighted.

Related pages

• eulers-number
• exponential-series
• logarithmic-series
• infinitesimal-and-infinite-numbers
• exponential-function
• logarithm
• change-of-base
• common-logarithm
• chapter-7-on-exponentials-and-logarithms-expressed-through-series