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Counting in two ways , Exponential functions and logarithms (other) , Integer and fractional parts. Archimedean property

Prove that for every natural number n $>$ 1 the equality: [$n^{1 / 2}] + [n^{1/ 3}] + … + [n^{1 / n}] = [log_{2}n] + [log_{3}n] + … + [log_{n}n]$ is satisfied.

Exponential functions and logarithms (other) , Rational and irrational numbers , The fundamental theorm of arithmetic. Prime factorisation.

For what natural numbers a and b is the number $log_{a}$b rational?

Exponential functions and logarithms (other) , Rational and irrational numbers , Surds. Rational powers (other)

Are there any irrational numbers a and b such that the degree of $a^b$ is a rational number?

Certain properties of a function and recurrence relations. , Exponential functions and logarithms (other)

The function f is such that for any positive x and y the equality f $($xy$)$ = f $($x$)$ + f $($y$)$ holds. Find f $($2007$)$ if f $($1/2007$)$ = 1.

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