   Chapter 11.4, Problem 42ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
1 views

# Show that ⌈ log 2 n ⌉ is Θ ( log 2 n ) .

To determine

To prove:

log2n is Θ(log2n)

Explanation

Given information:

log2n is Θ(log2n)

Definition used:

f is of order g:

f(x) is Θ(g(x)) if there exists a positive real numbers, A, B and a nonnegative real number k such that A|g(x)||f(x)|B|g(x)| whenever x>k

Ceiling function:

x smallest integer that is less than or equal to x.

Proof:

Since xx for all real numbers of x, we have |log2n ||log2n| whenever n0.

Since x1<xx,

|log2n ||log2n1|

| log2n ||log2nlog22|                       ( log22=1) =|log2n2|                       ( log2ab= log2a<

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