Logarithmic Curve

Summary: §§512–515. The curve $y=a{e}^{x/b}$, or equivalently $x=b\log (y/a)$, with constant subtangent $b$ everywhere. Arithmetic progression in $x$ corresponds to geometric progression in $y$. The negative-$x$ axis is an asymptote. Drawn as figure 101.

Sources: chapter21 §§512–515, figures99-102 (figure 101).

Last updated: 2026-05-12

---

Equation and basic shape (§§512–513)

Euler writes the equation in two equivalent forms:

$b\log \frac{y}{a}=x⟺y=a{e}^{x/b},$

where $\log$ denotes the natural logarithm (so $\log e=1$, $e=2.718281828\dots$) and the constant $b$ multiplies any chosen logarithm base into natural — making the chapter’s choice of base immaterial.

Geometric/arithmetic correspondence. Letting $e=m^n$ and $b=nc$, the equation becomes $y=a{m}^{x/c}$. Hence the table

$x$	$0$	$c$	$2c$	$3c$	$4c$	$\cdots$
$y$	$a$	$am$	$a{m}^2$	$a{m}^3$	$a{m}^4$	$\cdots$

extends to negative $x$ as $y=a/m,a/m^2,a/m^3,\dots$ — arithmetic progression on the abscissa produces geometric progression on the ordinate. This is the defining property.

Asymptotic behavior (§513, figure 101)

For positive $x$, $y$ increases monotonically to infinity. For negative $x$, $y\to 0$ but stays positive: the negative $x$-axis is an asymptote $Ap$ to the curve.

Reading off figure 101: the axis is horizontal, $A$ is the origin, $AB=a$ is the ordinate at $x=0$, and the curve passes through the points $(0,a),(c,am),(2c,a{m}^2),\dots ,(-c,a/m),(-2c,a/m^2),\dots$ The curve approaches the negative axis but never meets it.

Origin-invariance (§513)

The form $y=a{e}^{x/b}$ is preserved under translation of the origin. If $Aa=f$ and we choose $a$ as new origin (so $aP=t$ for $x=t-f$), then

$y=a{e}^{(t-f)/b}=\frac{a}{e^{f/b}}{e}^{t/b}=g{e}^{t/b}$

where $g=a/e^{f/b}$. The form is the same; only $g$ changes. Consequently

$\frac{aP}{b}=\log \frac{PM}{ab}.$

The constant $b$ (the logarithmic parameter) is intrinsic — it is the same in every choice of origin.

Constant subtangent (§§514–515)

The principal property of the logarithmic curve: the subtangent $PT$ has constant length $b$ everywhere.

Derivation. At a point $M=(x,y)$, take a nearby point $N=(x+u,y+\delta y)$ on the curve. Then

$QN-PM=a{e}^{x/b}\left(e^{u/b}-1\right)=a{e}^{x/b}\left(\frac{u}{b}+\frac{u^2}{2b^2}+\frac{u^3}{6b^3}+\cdots \right).$

The straight line $NM$ extended meets the axis at $T$, and similar triangles give $PT=u/(e^{u/b}-1)$, i.e.

$PT=\frac{1}{\frac{1}{b}+\frac{u}{2b^2}+\frac{u^2}{6b^3}+\cdots }.$

As $u\to 0$, the line $NM$ becomes the tangent at $M$, and $PT\to b$.

The logarithmic parameter $b$ is everywhere equal to the subtangent, which thus has constant length. (source: chapter21, §514)

This is the defining property in the classical pre-calculus literature: any curve with constant subtangent is the logarithmic curve.

Paradoxes (§515 onward)

The clean picture above is only the main branch $MBm$ of the logarithmic curve — the continuous one going to infinity in both directions and accumulating on the asymptote. Two paradoxes complicate matters:

• discrete-points-below-asymptote (§§515, 517). When $x/b$ has an even denominator, $y$ has two values, one positive and one negative. So below the asymptote there are infinitely many discrete points — dense but not continuous.
• infinite-logarithms-paradox (§516). Since $\log 1=0$ but $\log (-1{)}^2=2\log (-1)=0$, every number has infinitely many logarithms; only one is real. This connects to the previous paradox: the discrete points correspond to non-real logarithms made real by even-denominator root extraction.

Figures

Figures 99–102

Related pages

• chapter-21-on-transcendental-curves
• transcendental-curves
• discrete-points-below-asymptote — the parity anomaly on the same curve.
• infinite-logarithms-paradox — the multi-valued nature of $\log$.
• subtangent-and-subnormal — algebraic-curve version of this calculation (chapter 13).
• logarithmic-spiral — the polar cousin of this curve.