Like all logarithms, the natural logarithm maps multiplication of positive numbers into addition:
This definition therefore derives its own principal branch from the principal branch of nth roots.
How to establish this derivative of the natural logarithm depends on how it is defined firsthand. If the natural logarithm is defined as the integral
then the derivative immediately follows from the first part of the fundamental theorem of calculus.
Such techniques were used before calculators, by referring to numerical tables and performing manipulations such as those above.
These continued fractions—particularly the last—converge rapidly for values close to 1. However, the natural logarithms of much larger numbers can easily be computed, by repeatedly adding those of smaller numbers, with similarly rapid convergence.
The reciprocal of the natural logarithm can be also written in this way: