Exponent SimplificationI have this expression where u and k are arbitrary constants: u ( 1 k ) log k ⁡ ( log k ⁡ ( u ) ) I'm trying to clean up or simplify this expression... how can i go just making this cleaner? I'm forgetting log and exponent rules. Thanks!

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Exponent SimplificationI have this expression where u and k are arbitrary constants:            u                                    (                                          1                k                                      )                                              log              k                        ⁡            (                          log              k                        ⁡            (            u            )            )                              I&#039;m trying to clean up or simplify this expression... how can i go just making this cleaner? I&#039;m forgetting log and exponent rules. Thanks!

na1p1a2pafr · Accepted Answer

Note that            (              1        k            )                      log        k            ⁡      (              log        k            ⁡      (      u      )      )        =      1          k                        log          k                ⁡        (                  log          k                ⁡        (        u        )        )            Since       k                  log        k            ⁡      (      a      )        =  a, then      1          k                        log          k                ⁡        (                  log          k                ⁡        (        u        )        )              =      1                  log        k            ⁡      (      u      )      Since   1  =      log    k    ⁡  (  k  ) and by change of base formula                     log        b            ⁡      (      p      )                      log        b            ⁡      (      q      )        =      log    q    ⁡  (  p  ), then      1                  log        k            ⁡      (      u      )        =                    log        k            ⁡      (      k      )                      log        k            ⁡      (      u      )        =      log    u    ⁡  (  k  )Recalling once again that       u                  log        u            ⁡      (      k      )        =  k      u                            (                      1            k                    )                                      log            k                    ⁡          (                      log            k                    ⁡          (          u          )          )                      =      u                  log        u            ⁡      (      k      )        =  k

Bairaxx · Accepted Answer

(      1    k        )                  log                  k                    ⁡      (              log                  k                    ⁡      (      u      )      )        =      k          −              log                  k                    ⁡      (              log                  k                    ⁡      (      u      )      )        =  =      k                  log                  k                    ⁡      (              1                              log                          k                                ⁡          (          u          )                    )        =      1                  log                  k                    ⁡      (      u      )        =      log          u        ⁡  (  k  )then      u          (              1        k                    )                              log                          k                                ⁡          (                      log                          k                                ⁡          (          u          )          )                      =  k

I have this expression where u and k are arbitrary constants: u^(k(log_k(log_k(u)))) I'm trying to clean up or simplify this expression... how can i go just making this cleaner? I'm forgetting log and exponent rules. Thanks!

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