I have a bit of a unique problem. Well, maybe not a problem because I'm really just curious about it, but...
I have a simple 8 digit calculator. It has +, -, x, /, and a constant operation function. It has no usable memory slot.
I need (lit. would like to know how) to calculate ln(x) on this calculator. Any series I have found lose a lot of precision since whatever is outside the 8 digit range is truncated without rounding. I have used Padé approximants with trig functions, and a continued fraction for exp(), but none of these two methods are adequate for ln(x), either because there is no way around the truncation error or it requires writing down many intermediate values, or it requires memorizing and punching in many constants.
Does anyone know any good way to calculate ln(x) (or any other base, since memorizing one constant is OK) to as many digits as possible as easily as possible on this calculator (bonus points to methods where the value on the screen is immediately, meaning you don't need to write it down)?
For that matter, if you happen to know some good ways to calculate other special functions, I am interested in that as well.