It is easy to show log isn't a polynomial (no continuous extension to R )....





It is easy to show log isn't a polynomial (no continuous extension to R ). More challenging is showing it isn't rational.
Suppose it were a rational function. Then write, the fraction in lowest terms
log x = G ( x ) Q ( x ) G ( x ) log x = Q ( x )
Clearly, as x 0, Q ( x ) 0. However, Q ( x ) then has x as factor so that
G ( x ) x log x = Q 2 ( x )
It is well known that x log x 0 as x 0, so for Q 2 ( x ) to have a finite limit as x 0, which it must since it is a polynomial, G ( x ) 0 as x 0 so that x is a factor of G ( x ). This contradicts the assumption that G Q was in lowest terms.
If there is something wrong with this, please comment, but my main question is
What are some other ways to prove that logx isn't a rational function?

Answer & Explanation




2022-07-13Added 15 answers

One reason is that a rational function is defined over all R except for a finite number of points, but log is not.
Let's prove that log is not even a rational function restricted to ( 0 , + ).
If log = G Q , then 1 x = log = G Q G Q Q 2 .
This implies that G and Q have the same degree and therefore lim x G ( x ) Q ( x ) is finite.
But lim x log ( x ) = .



2022-07-14Added 6 answers

A rational function r vanishes at or there is an integer power q so that r ( x ) c x q as x . The log function exhibits none of these behaviors.

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