Natural Logarithm of Ten

ln 10 = 2.302585 is the bridge between the two logarithm bases that matter: ln x = 2.302585 × log₁₀ x. It shows up wherever a base-10 quantity is differentiated, most famously inside the Nernst equation, where the group 2.303·RT/F evaluates to 0.05916 V at 25 °C and gives the familiar 59.16 mV per decade slope of a pH or ion-selective electrode.

It plays the same role in chemical kinetics, where converting an Arrhenius plot from ln k to log k multiplies the slope by 2.303, and in acoustics and control theory, where a decade and a neper are related by the same factor. Before calculators, tables were published in base 10 because the characteristic and mantissa split made multiplication easy on paper, while the calculus wanted base e — so 2.303 was the number every engineer kept in their head to move between the two worlds.