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Exponentials and logarithms in prime characteristic

Taking the exponential of a derivation is a classical tool for producing an automorphism of an algebra, under suitable conditions. Over fields of positive characteristic modified exponentials such as a truncated exponential have been used for decades, notably in the technique of toral switching, a fundamental tool for the study of modular Lie algebras. A crucial aspect of that technique is that such modified exponentials of derivations need not be automorphisms, but they still turn a particular grading of the algebra into another grading. I will review such modified exponentials of various generality, focussing on a weak functional equation which makes their application to gradings work. I will then introduce similar modified logarithms and discuss some of their properties.