# Answered Essay: Order the following functions by the most accurate asymptotic not

Order the following functions by the most accurate asymptotic notation you learned in this course.

n!, n^2 + √ n (log n)^10, n^1/3 , (log n)^100, n ^3 , 2^n , 10^( √ n ) , 2 ^ (log n) , 2 ^(2 log n) , 2 ^(√ log n) , 128, 128n.

Indicate which functions grow at the same rate (all logarithms are base 2). For example, if you are asked to order n, 2n and 2n^2 , then you answer should be “n = Θ(2n), 2n = o(2n^2 )”.

Please pay attention to the functions like 10^( √ n ) since other answers take it as 10√ n which is not what I want. Thanks

128, (log n)^100, 2 ^(√ log n) , n^1/3 , 2 ^ (log n) , 128n, 2 ^(2 log n) , n^2 + √ n (log n)^10, n ^3 , 10^( √ n ) , 2^n , n!

And
2 ^ (log n) =$\theta ($ 128n)
2 ^(2 log n) =$\theta ($ n^2 + √ n (log n)^10)

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