Give the first ten terms of the following sequences. You can assume that the sequences start with an index of 1. Logs are to base 2.

(a) The n term is [log n]

(b) The n term is 2^ (log n)

a)

a1 = log (1) = 0 (2⁰ = 1)

a2 = log (2) = 1 (2¹ = 2)

a3 = log (3) = ln (3) / ln (2) = 1.098/0.693 = 1.5849

a4 = log (4) = 2 (2² = 4)

a5 = log (5) = ln (5) / ln (2) = 1.610/0.693 = 2.322

a6 = log (6) = log (3*2) = log (3) + log (2) = 1.5849+1 = 2.5849 (here I use the property log (a*b) = log (a) + log (b)

a7 = log (7) = ln (7) / ln (2) = 1.9459/0.6932 = 2.807

a8 = log (8) = 3 (2³ = 8)

a9 = log (9) = log (3²) = 2*log (3) = 2*1.5849 = 3.1699 (I use the property log (a^k) = k*log (a))

a10 = log (10) = log (2*5) = log (2) + log (5) = 1 + 2.322 = 3.322

b) I can take the results of log n we previously computed above to calculate 2^log (n), however the idea of this exercise is to learn about the definition of log_2:

log (x) is the number L such that 2^L = x. Therefore 2^log (n) = n if we take the log in base 2. This means that

a1 = 1

a2 = 2

a3 = 3

a4 = 4

a5 = 5

a6 = 6

a7 = 7

a8 = 8

a9 = 9

a10 = 10

I hope this works for you!