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 isWhat are some other ways to prove that logx isn't a rational function?

Question

It is easy to show   log isn&#039;t a polynomial (no continuous extension to       R  ). More challenging is showing it isn&#039;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 isWhat are some other ways to prove that logx isn&#039;t a rational function?

pampatsha · Accepted Answer

One reason is that a rational function is defined over all       R   except for a finite number of points, but   log is not.Let&#039;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    )    =    ∞  .

civilnogwu · Accepted Answer

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.