ln() vs. log()

[Dieter]

(04-19-2016 02:00 PM)jebem Wrote:  [ ln 10000 : ln 10 = 4,000 0000 0001 ]
However, taking the popular top class Texas ti-89 Titanium (at least in other forums...), with a setup to display 12 float digits in scientific format, it returns Zero for both cases.
Is this impressive?

No, it's simply wrong.

ln 10000, evaluated to 12 digits, equals 9,210 3403 7198.
ln 10, evaluated to 12 digits, equals 2,302 5850 9299.

The quotient of both values is 4,000 0000 0000 868... which is 4,000 0000 0001 when rounded to the same 12 digits again. So the exact result of this quotient is not (!) exactly 4. If you get a plain 4 the calculator is doing something behind your back you do not know.

Now do the same calculation with merely 10 digits. Here the 10-digit representations of ln 10000 resp. ln 10 happen to match the exact value quite closely:

ln 10000, evaluated to 10 digits, equals 9,210 3403 72.
ln 10, evaluated to 10 digits, equals 2,302 5850 93.

The quotient of both values is exactly 4. This is not because of more accuracy but simply because the two intermediate values happen to round to 10 digits they way they do. Which in turn is caused by the fact that ln 10 happens to round so nicely to 10 digits.

Dieter