For the number z = i i i can I take a log on both sides and write it as log ⁡ ( z ) = i i log ⁡ ( i )?I know that we can write log ⁡ ( e i θ ) = i θ log ⁡ ( e ), but I'm no sure if we can do that when the base is non real. Is this step valid for a non real base?

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For the number  z  =      i                  i        i            can I take a log on both sides and write it as   log  ⁡  (  z  )  =      i          i        log  ⁡  (  i  )?I know that we can write   log  ⁡  (      e          i              θ              )  =  i      θ    log  ⁡  (  e  ), but I&#039;m no sure if we can do that when the base is non real. Is this step valid for a non real base?

Siena Bennett · Accepted Answer

I don&#039;t think that will work. But you could simplify is by using the definition from top to down. First, compute   w  =      i    i   and then   z  =      i    w  By definition      i    i    =      e          i      log      ⁡      (      i      )      where   log  ⁡  (  i  )  =  ln  ⁡  (      |    i      |    )  +  A  r  g  (  i  )  i  =      1    2    π  i. So we get   w  =      i    i    =      e          −              1        2            π      . Next,  z  =      i    w    =      e          w      log      ⁡      (      i      )        =      e                  1        2            w      π      i        =      e          i              1        2            π              e                  −                      1            2                    π                      .

For the number z=i^(i^i) can I take a log on both sides and write it as log(z)=i^i log(i)?

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New Questions in Calculus 1