Exponential Curves

Summary: §518. Curves with a variable in the exponent, exemplified by $y=x^x$. Treated as belonging to the logarithmic genus because $m\log y=x\log x$. Minimum at $x=1/e$. Discrete points below the axis for negative $x$ (the same parity anomaly as the logarithmic-curve). Drawn as figure 102.

Sources: chapter21 §518, figures99-102 (figure 102).

Last updated: 2026-05-12

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The equation (§518)

The simplest exponential curve is

$y=x^x,$

equivalently $m\log y=x\log x$ for any logarithm base. Because the equation can be put into logarithmic form, it falls under the genus logarithmic curves — Euler assigns “any equation in which not only logarithms, but also any variable exponent occur” to that genus.

Shape on the positive axis (§518, figure 102)

Sample values:

$x$	$y=x^x$
$0$	$1$ (limit)
$1/e\approx 0.368$	$\approx 0.692$ (minimum)
$1/2$	$\sqrt{1/2}\approx 0.707$
$1$	$1$
$2$	$4$
$3$	$27$
$⋮$	$⋮$

Reading figure 102: axis $AP$ to the right, vertex at $B$ on the positive axis, $AC=1,AB=1$. Between $A$ and $C$ ($0<x<1$) the curve $BDM$ dips below height $1$, reaching its minimum at $D=(1/e,e^{-1/e})=(0.368,0.692)$. From $D$ outward the curve climbs steeply to infinity.

The minimum value of the ordinate occurs when the abscissa $x=1/e=0.36787944$ and then the ordinate $y=0.6922005$, as we shall show in what follows. (source: chapter21, §518)

The “as we shall show” is a forward reference to differential calculus — set $\frac{d}{dx}(x^x)=x^x(\log x+1)=0$ gives $\log x=-1$, i.e. $x=1/e$.

Negative $x$: discrete points (§518)

For $x<0$, write $y=1/(-x{)}^x$. The same parity rule as the logarithmic-curve applies:

• $x$ an integer or fraction with odd denominator: one real value of $y$ (positive if $x$ is even, negative if odd).
• $x$ a fraction with even denominator: two real values $\pm \sqrt{|(-x{)}^x|}$.
• Otherwise: complex.

Hence the negative-$x$ part of $y=x^x$ is a scatter of discrete points on both sides of the axis, dense but not contiguous, asymptotically converging to the axis — the same discrete-points-below-asymptote phenomenon. The continuous part on the positive axis “terminates abruptly at $B$” without algebraic continuation, in apparent violation of the law of continuity — Euler defuses this by noting the discrete points fill in the missing branch in a generalized sense.

Relation to §511

The §511 complex-exponent curve $2y=x^i+x^{-i}=2\cos \log x$ is also “exponential” — variable in exponent — but produces a bounded oscillating real curve. The same genus assignment applies.

Figures

Figures 99–102

Related pages

• chapter-21-on-transcendental-curves
• logarithmic-curve — host genus.
• discrete-points-below-asymptote — the same anomaly on the negative axis.
• implicit-exponential-curve — §519’s $z^y=y^z$, where both $x$ and $y$ appear in exponents.
• transcendental-curves