To begin note that 2^{10} = 1024 which is very close to 1000 = 10^{3}.
2^{10} ≈ 10^{3}
10 log 2 ≈ 3
log 2 ≈ 3/10 = 12/40
For 3, note that 3^{4} = 81 which is very close to 80.
3^{4} ≈ 80 = 10*2^{3}
4 log 3 ≈ 1 + 3 log2 ≈ 76/40
log 3 ≈ 19/40
Since 4 = 2^{2}, log 4 = 2 log 2 ≈ 2*12/40 = 24/40
Since 5 = 10/2, log 5 = log 10 - log 2 ≈ 1 - 12/40 = 28/40
Since 6 = 2*3, log 6 = log 2 + log 3 ≈ 12/40 + 19/40 = 31/40
For 7, note 7^{2} = 49 which is close to 50 = 100/2
log 7^{2} = log 100/2
2 log 7 = 2 - log 2 ≈ 2 - 12/40 = 68/40
log 7 ≈ 34/40,
see the comments for an argument about why 33/40 is also pretty good.
Since 8 = 2^{3}, log 8 = 3 log 2 ≈ 3*12/40=36/40
Since 9 = 3^{3}, log 9 = 2 log 3 ≈ 2*19/40=38/40 |