Actually, now that I look at this, I see a beautiful parallel to the logarithm that I had previously thought was missing.
As we move away from the origin, the logarithm grows linearly as the distance from the origin grows exponentially. In parallel fashion, as we move away from the origin, the superlogarithm grows linearly as the distance from the origin grows superexponentially.
Going the other way, as we approach the origin, the logarithm decreases linearly as the distance from the origin decreases exponentially. However, the logarithm decreases linearly as the distance from the origin decreases linearly.
For example, log(0.049787) is about -3, but slog(0.049787) is about -0.955, give or take. Moving in further, log(0.0000454) is about -10, but the slog is about -0.99996.
However, if we look in the primary exponential branch of the slog, and loop around the upper primary singularity, we'll see a singularity at the origin. As we approach this singularity, the value of the slog increases linearly as the distance from the origin decreases superexponentially! (By "decrease superexponentially", I mean the reciprocal increases superexponentially.)
The branches of the superlogarithm provide many layers of complexity, something I assume would be worthy to be explored in a Ph.D. thesis. Alas, it'll be years before I could dream of embarking on such an adventure, but it's still exciting to be exploring this relatively new territory.
As we move away from the origin, the logarithm grows linearly as the distance from the origin grows exponentially. In parallel fashion, as we move away from the origin, the superlogarithm grows linearly as the distance from the origin grows superexponentially.
Going the other way, as we approach the origin, the logarithm decreases linearly as the distance from the origin decreases exponentially. However, the logarithm decreases linearly as the distance from the origin decreases linearly.
For example, log(0.049787) is about -3, but slog(0.049787) is about -0.955, give or take. Moving in further, log(0.0000454) is about -10, but the slog is about -0.99996.
However, if we look in the primary exponential branch of the slog, and loop around the upper primary singularity, we'll see a singularity at the origin. As we approach this singularity, the value of the slog increases linearly as the distance from the origin decreases superexponentially! (By "decrease superexponentially", I mean the reciprocal increases superexponentially.)
The branches of the superlogarithm provide many layers of complexity, something I assume would be worthy to be explored in a Ph.D. thesis. Alas, it'll be years before I could dream of embarking on such an adventure, but it's still exciting to be exploring this relatively new territory.
~ Jay Daniel Fox
