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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 $\exp(x)= 5$.

  1. What is $x$? rational , irrational?

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

  3. Can a combination of irrational number result into a rational number?

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I don't really know what the question is, but $\ln(5)$ is irrational, a consequence of the fact that $e$ is transcendental. If $e^{p/q}=5$, then $e^p=5^q$. Slightly more generally, if $e^x$ is rational, then $x$ is irrational. Related question where you'll find a much more general statement:… – Jonas Meyer Jan 11 '13 at 6:22
What @Jonas said. Regarding Can a combination of irrational number(s) ... result into a rational number?, consider $\pi+(42-\pi)$. – Did Jan 11 '13 at 6:30
$(1/2 + \pi) + (1/2 - \pi) = 1$. – mjqxxxx Jan 11 '13 at 6:30
@Jonas, also $x$ must be nonzero. – sdcvvc Jan 11 '13 at 6:58
Also, if x is rational, then e^x is irrational. This statement is different to Jonas'. – Adam Rubinson Jan 11 '13 at 8:53

(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), And any rational can be written as the ratio of two integers.

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What precisely do you mean by a "combination of irrational number[s]"?

Obviously $\sqrt{2}$ and $-\sqrt{2}$ are both irrational, and if "combining" them allows for the operation of addition, then just observe the sum is the rational number $0$.

If you are more interested in questions surrounding exponentiation, logarithms, and the such, you might check out this recent MO question in a similar spirit.

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Also Rationals are dense in Reals and Irrationals are dense in reals

which implies that you

  1. can always find a sequence of irrational numbers that converges to a given rational

  2. can always find a sequence of rational numbers that converges to a given irrational

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