Can a rational number be represented by a combination of irrational numbers? Any real number is some power of $e$ (because $\ln(x)$ has values in the range $(-\infty , + \infty)$.

Say, $5$ is a rational number. So there is some $x$ which makes $\exp(x)= 5$.

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*What is $x$? rational , irrational?


*A power of $e$  leads to $5$, Will it be an infinite series converging to the value $5$?


*Can a combination of irrational number result into a rational number?


(When he was 12) George Bergman proved that any integer can be represented by a finite sum of integral powers of the golden ratio (irrational number), http://en.wikipedia.org/wiki/Golden_ratio_base.  And any rational can be written as the ratio of two integers.
