For each positive real number Use this fact and the result of exercise 21 in Section 11.1 to prove the following: For every integer , if x is any real number with then .
For all integers n = 1, log2x < x1/nfor all real numbers x >(2n)2n.
For all positive real numbers u, log2u < u.
Let n be a positive integer and let x be a positive real number such that
Since x is a positive real number, is also a positive real number.
We know that the inequality holds for all positive real numbers.
Multiply each side of the equation by 2n (note that n > 0)
However, we also know that Let us take the 1/2th power of each side (note that is an increasing function for all positive integersm and n ).
Multiply both sides by
Take the 1/ nth power of each side (note that is an increasing function for all positive integersm and n )
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