Noncircular construction of e and ln for the real lineCould anyone direct me to (or possibly detail) a construction of e and ln along the reals?For example, they can define e = lim n → ∞ ( 1 + 1 n ) n but from this definition how do they prove:It converges! d d x e x = e x etc.!Then if we know e x injective from R → R + , we can call ln ⁡ ( x ) the inverse of it. If we can prove ln is differentiable on its domain, then we can say: 1 = d d x x = d d x e ln ⁡ ( x ) = e ln ⁡ ( x ) ⋅ ln ⁡ ( x ) ′ = x ⋅ ln ⁡ ( x ) ⇒ ln ⁡ ( x ) ′ = x − 1 but this all depends on the above.

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Noncircular construction of   e and   ln for the real lineCould anyone direct me to (or possibly detail) a construction of   e and   ln along the reals?For example, they can define   e  =      lim          n      →      ∞        (  1  +      1    n        )    n   but from this definition how do they prove:It converges!      d          d      x            e    x    =      e    x  etc.!Then if we know       e    x   injective from       R    →            R      +      , we can call   ln  ⁡  (  x  ) the inverse of it. If we can prove   ln is differentiable on its domain, then we can say:  1  =      d          d      x        x  =      d          d      x            e          ln      ⁡      (      x      )        =      e          ln      ⁡      (      x      )        ⋅  ln  ⁡  (  x      )    ′    =  x  ⋅  ln  ⁡  (  x  )  ⇒  ln  ⁡  (  x      )    ′    =      x          −      1      but this all depends on the above.

Norah Baxter · Accepted Answer

There are many possible paths through the exposition graph.select a definition one or both of       e    x   and   log  ⁡  x for real   xselect a definition of   e prove all standard compatibilities between these definitions (such as   e  =  exp  ⁡  (  1  )  =      log          −      1        ⁡  (  1  )  =  lim  (  1  +      1    x        )    x    =  ∑      1          n      !       ), equivalence to other standard definitions, and basic properties of the functions such as functional equations and power series developments.Almost every book on real analysis, or on &quot;rigorous calculus&quot; (Spivak, Apostol or similar), and many ordinary calculus books that do not claim complete rigor but achieve it in this part of the exposition, will choose at least one path through the maze that accomplishes what you want.

skottyrottenmf · Accepted Answer

We can define   l  n  (  x  )  =      ∫          1              x                  d      t        t   and define       e    x   as its inverse.However, even the apprach you suggested works. Showing that the above sequence converges is not very hard. You can show that the sequence is increasing.Also, you can show that the &quot;power series&quot; for       e    x   converges absolutely.

Noncircular construction of e and ln for the real line Could anyone direct me to (or poss

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