This thread was about logs, as I recall. Now the stam log is actually the natural log. It is the only one with the magic properties that the exponential has, which is that no matter how many times you differentiate it, it remains the same function.
Some books will write it as log, although others use ln. Matlab will use log10 for the base 10 log function. In complex variables, I think they use log or Log depending on whether it is the single valued or multi-valued function.
Anyway, regarding the magic property of the exponential, the Thomas text on Calculus has this mashal to try to explain it:
A student once came to the guru and asked, what holds up the world. The guru thought for a long time and said, an elephant. The student then asked, well, what holds up the elephant? The guru thought about it, and replied, another elephant. The student then asked, well, what holds up that elephant? The guru thought again and responded, it’s elephants all the way down.