The infinite log operation:

Consider taking the logarithm of a number “A” to a base “B”.

Consider the following problem where “x” is known and “y” is sought or where “y” is known and “x” is sought:

Finding “x” for a given “y” simply involves raising “y” to the power “1/y”. Finding “y” for a given “x” is more complicated. The use of the Lambert Function as described above is inconvenient, but there is another way to solve for “y” given “x” (real or complex). The easiest way to get the eigenlog “y” for an arbitrary value of the eigenbase “x” is via taking an “infinite number of logs”. Of course, one does not actually take the log an infinite number of times, but rather the log is taken until the value of the calculated eigenlog converges to an acceptable degree of accuracy. Note that the eigenlog can be multivalued for a given eigenbase.

For a more detailed description, see the paper linked below:

Tetration/Exponentiation - the infinite log operation