Then, using the identity R 2 v e c = exp ⁡ ⁡ [ 2 v e c ln ⁡ ⁡ R ] (here "exp" denotes the exponential function ; see also Natural logarithm as well as the "power" identity at Logarithmic identities) and the identity tanh ⁡ ⁡ x = e 2 x − − 1 e 2 x + 1 (see Hyperbolic function), this is equivalent to Δ Δ v = c tanh ⁡ ⁡ ( v e c ln ⁡ ⁡ m 0 m 1 )