# §4.12 Generalized Logarithms and Exponentials

A generalized exponential function satisfies the equations

and is strictly increasing when . Its inverse is called a generalized logarithm. It, too, is strictly increasing when , and

These functions are not unique. The simplest choice is given by

Then

and

where the exponentiations are carried out times. Correspondingly,

and

where denotes the -th repeated logarithm of , and is the positive integer determined by the condition

4.12.10

Both and are continuously differentiable.

For further information, see Clenshaw et al. (1986). For generalized logarithms, see Walker (1991). For analytic generalized logarithms, see Kneser (1950).