Mistakes: Where is Mathematica's or my mistake

Questions
1

Hello,

entering the following commands

N[Log[10^3]/Log[10], 20]
N[Log[10^3]/Log[10] + 1., 20]
Floor[%]

in mathematica 8 I get the following output:

3.0000
4.
3

I think, of course, the last output should be 4 and not 3.
Otherwise one would have Floor[4.]=3 and this is nonsense.

In consequence there is an additional similiar problem.
Entering
Floor@(Log[10, 10^Range[10]] + 1)
Floor@(Log[10^Range[10]]/Log[10] + 1.)

one gets as output
{2, 3, 4, 5, 6, 7, 8, 9, 10, 11}
{2, 3, 3, 5, 5, 6, 7, 9, 9, 10}

My students and me were very astonished and could not understand
mathematica outputting such obvious nonsense (the last output agaim).
Of course, both outputlines should be identical.

Could someone in the community give me an advice?
Thanks in advance!

Peter Quedens

2

This is a good opportunity to teach your students about how numbers (real and integer) are stored in a computer and displayed on the screen. The function RealDigits will be helpful:

RealDigits[Log[10^3]/Log[10] + 1.]
RealDigits[N[Log[10^3]/Log[10] + 1., 20]]