To find the Legendre transformation of the Legendre transformation of f , f ∗ ∗ ∗ ∗ ( x ) = sup x ∗ ∗ ∈ ∈ R ( x x ∗ ∗ − − x ∗ ∗ ( ln ⁡ ⁡ ( x ∗ ∗ ) − − 1 ) ) , x ∈ ∈ I , where a variable x is intentionally used as the argument of the function f ∗ ∗ ∗ ∗ to show the involution property of the Legendre transform as f ∗ ∗ ∗ ∗ = f . we compute 0 = d d x ∗ ∗ ( x x ∗ ∗ − − x ∗ ∗ ( ln ⁡ ⁡ ( x ∗ ∗ ) − − 1 ) ) = x − − ln ⁡ ⁡ ( x ∗ ∗ ) thus the maximum occurs at x ∗ ∗ = e x because the second derivative d 2 d x ∗ ∗ 2 f ∗ ∗ ∗ ∗ ( x ) = − − 1 x ∗ ∗ < 0 over the domain of f ∗ ∗ ∗ ∗ as I ∗ ∗ = ( 0 , ∞ ∞ ) . As a result, f ∗ ∗ ∗ ∗ is found as f ∗ ∗ ∗ ∗ ( x ) = x e x − − e x ( ln ⁡ ⁡ ( e x ) − − 1 ) = e x , thereby confirming that f = f ∗ ∗ ∗ ∗ , as expected.