For each positive real number
To prove:
For all integers n = 1, log2x < x1/nfor all real numbers x >(2n)2n.
Given information:
For all positive real numbers u, log2u < u.
Proof:
Let n be a positive integer and let x be a positive real number such that
Since x is a positive real number,
We know that the inequality
Let
Multiply each side of the equation by 2n (note that n > 0)
Property logarithm:
However, we also know that
Multiply both sides by
Note that
Take the 1/ nth power of each side (note that