08/30/2007, 11:08 PM

From a practical standpoint, I've considered the solution to bases as being exponentially scaled sine waves of period 2 as we approach positive infinity, with iterated logarithms to get us back to the origin. This is in line with the solutions for bases between 1 and eta, which are essentially exponential near positive infinity, with iterated logarithms getting us back to the origin. For b=e^-e, we wouldn't even need an exponential scaling, just a plain old sine wave (near x=infinity), just as with eta, which becomes essentially linear near positive infinity.

For solutions less than b=e^-e, we wouldn't even need an infinitesimal sine wave. A sine wave oscillating between the upper and lower point would make the most sense, with an infinitesimal perturbance which, with an infinite iteration of logarithms, would gets us back to the origin. Limits would have to be taken, of course, but it seems that a plain old sine wave would make the most sense, barring any solid evidence that the oscillations are somehow more complex than the problem would seem to dictate.

For solutions less than b=e^-e, we wouldn't even need an infinitesimal sine wave. A sine wave oscillating between the upper and lower point would make the most sense, with an infinitesimal perturbance which, with an infinite iteration of logarithms, would gets us back to the origin. Limits would have to be taken, of course, but it seems that a plain old sine wave would make the most sense, barring any solid evidence that the oscillations are somehow more complex than the problem would seem to dictate.

~ Jay Daniel Fox