Can a^x be negative infinity

  1. I think about on this and found a result, [tex]e^{i\pi k}=e^{-i\pi k}=-1\Rightarrow i\pi k=ln(-1)[/tex]
    a>1 and k=1,2,3...

    [tex]y=(-a)^{i\alpha }\Rightarrow lny=i\alpha ln(-a)=i\alpha (ln(-1)+lna)=i\alpha (i\pi k+lna)\Rightarrow y=e^{-\alpha \pi k}e^{i\alpha \ln{a}}[/tex]

    Then

    [tex]\alpha \ln{a}=-\pi \Rightarrow \alpha =\frac{-\pi }{\ln{a}}\Rightarrow e^{-i\pi }e^{\frac{{\pi }^2 k}{\ln{a}}}=-e^{\frac{{\pi }^2 k}{\ln{a}}} [/tex]

    If I choose k as infinity, I get a exponential expression as negative infinity.
    Is it right? Please explain to me. Thanks
     
    Last edited: Sep 16, 2010
  2. jcsd
  3. Sorry, k would be k=2n-1=1,3,5,7...
     
  4. Not sure what you're doing, but I'd write:
    [tex]a^x=-k[/tex]

    [tex]e^{x\log(a)}=-k[/tex]

    [tex]x\log(a)=\log(-k)[/tex]

    [tex]x=\frac{\log(-k)}{\log(a)}=\frac{\ln(k)+i(\pi+2 n \pi)}{\log(a)}[/tex]

    and as [itex]k\to\infty[/itex], [itex]x\to \infty+i(\pi+2n\pi)/\log(a)[/itex]
     
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