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Showing content from https://dlmf.nist.gov/4.12 below:

§4.12 Generalized Logarithms and Exponentials ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions

§4.12 Generalized Logarithms and Exponentials

A generalized exponential function ϕ ⁡ ( x ) satisfies the equations

and is strictly increasing when 0 ≤ x ≤ 1 . Its inverse ψ ⁡ ( x ) is called a generalized logarithm. It, too, is strictly increasing when 0 ≤ x ≤ 1 , and

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

Then

and

Correspondingly,

and

where ℓ is the positive integer determined by the condition

Both ϕ ⁡ ( x ) and ψ ⁡ ( x ) are continuously differentiable.

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


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