Let x=2^100,〖y=3〗^60 and z=〖10〗^30 . What is the smallest number among the three?
x = 2^100
log x = 100 log 2
y = 3^60
log y = 60 log 3
z = 10^30
log z = 30 log 10
which log is lightest?
I read that as:
x = 2^100 , y = 3^60 , z = 10^30
take log of each one ...
logx = 100log2 = appr 30.103
logy = 60log3 = appr 28.63
logz = 30log10 = 30
logy < logz < logx
y < z < x
so 3^60 is the smallest
(my calculator was actually able to find the values in scientific notation, we could have decided that way)
To find the smallest number among the three, we need to compare their values.
First, let's calculate the values of x, y, and z:
x = 2^100 = 1,267,650,600,228,229,401,496,703,205,376
y = 3^60 = 4,782,969,289,640,000
z = 10^30 = 1,000,000,000,000,000,000,000,000,000
Now we can compare the values:
x < y since 1,267,650,600,228,229,401,496,703,205,376 < 4,782,969,289,640,000
x < z since 1,267,650,600,228,229,401,496,703,205,376 < 1,000,000,000,000,000,000,000,000,000
Therefore, the smallest number among the three is x = 1,267,650,600,228,229,401,496,703,205,376.