BIG1=63^9
BIG2=33^11
Which of the BIGs is bigger?
No calculating aids, just p&p
I keep a minimal set of logarithms in my head:
log(2)= .30
log(3)= .48
log(4)= .60 (twice that of 2)
log(5)= .70 (log(10) - log(2))
log(6)= .78 .30 + .48
log(7)= .85
log(8)= .90 3*.30
log(9)= .95 interpolation between .9 and 1
with log(7) the last to be remembered, basically as an interpolation between those of 6 and of 8, erring on the high side.
The memorization of the key items, .30 and .48, is made easier by visualizing a slide rule, and a graffito on the wall of Columbia University at 3010 Broadway: log(2) = .3010, about 50 years ago.
63 is 3*3*7 or 9*7 so I'll compromise between
2*.48 + .85 = 1.81 and
.95 + .85 = 1.8
Multiply by 9 to get 16.245.
For 33, I could extrapolate to get log(11)=1.05 and add .48 getting 1.53, or I could try log(100) - .48 = 2-.48 = 1.52. Multiplying by 11 gives 16.83 and 16.72 respectively. Either way it's bigger than 16.245.
BIG2 is bigger.
Check via calculator:
>> vpa(63^9)
ans =
15633814156853824.0
>> log10(ans)
ans =
16.194064945082235375499625026555
>> vpa(33^11)
ans =
50542106513726816.0
>> log10(ans)
ans =
16.703653338656762249904780438087
So, while the log estimate was a little high in each case, they were close enough to see which was bigger.
|
|
Posted by Charlie
on 2023-07-03 08:56:30 |
a solution
