How to represent 1/log(x) as integral of exponentials on domain [ 1 , ∞ )?

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How to represent 1/log(x) as integral of exponentials on domain   [  1  ,  ∞  )?

Alannah Hanson · Accepted Answer

Not sure if this is what you&#039;re looking for, but you can do something like this: let   x  &amp;gt;  a  &amp;gt;  1 be fixed, then by the fundamental theorem of calculus we have                              1                      log            ⁡            x                                              =                  1                      log            ⁡            a                          +                  ∫          a          x                                      (                                    1                      log            ⁡            t                                                              )                                ′                         d        t                                          =                  1                      log            ⁡            a                          +                  ∫          a          x                                      −            1                                              log              2                        ⁡            t                          ⋅                  1          t                         d        t            Next we can introduce a substitution   u  =  log  ⁡  (  log  ⁡  t  ), equivalently       e    u    =  log  ⁡  t which further implies       e    u       d  u  =      1    t       d  t. We then have                              1                      log            ⁡            x                                              =                  1                      log            ⁡            a                          +                  ∫                      log            ⁡            (            log            ⁡            a            )                                log            ⁡            (            log            ⁡            x            )                                                −            1                                (                          e              u                                      )              2                                      ⋅                  e          u                         d        u                                          =                  1                      log            ⁡            a                          −                  ∫                      log            ⁡            (            log            ⁡            a            )                                log            ⁡            (            log            ⁡            x            )                                    e                      −            u                                   d        u

How to represent 1/log(x) as integral of exponentials on domain [1, infty)?

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