aˣ vs xᵃ — Intersection Explorer

Parameter

a (base) 80.00

Main Plot View

Derivation

a^x = x^a

⇔ x · ln(a) = a · ln(x)

⇔ ln(x)/x = ln(a)/a

f(t) = ln(t)/t peaks at t = e with value 1/e ≈ 0.368. For each level below the max, two values of t hit it — that's the twin.

Twin Location Rule

a > e: twin ∈ (1, e)
1 < a < e: twin ∈ (e, ∞)
a = e: tangent — no second crossing

Famous Case

2⁴ = 4² = 16
The only integer pair with a ≠ x.

y = a^x

y = x^a

intersection

hover plot for values

f(t) = ln(t)/t — geometric meaning of the twin

Intersections in Q1

Twin (non-trivial) —

Trivial (x = a) —

—

Values at Twin

a^x_twin —

x_twin^a —

Values at x = a

a^a —

Diagnostic (ln space)

ln(a) / a —

1/e (max of f) 0.367879

Parametric Form

Setting r = x_twin / a, all twin pairs satisfy:

a = r^1/(r-1)

x = r^r/(r-1)

current r —

At your x₁ = 15.3

80^15.3 1.310 × 10²⁹

15.3⁸⁰ 5.961 × 10⁹⁴

15.3 is between the twin (≈1.06) and trivial (80) intersections, so x^a dominates by ~65 orders of magnitude.

ax ⋂ xa — Intersection Explorer