e ln ⁡ ( − 2 ) = − 2 but ln ⁡ ( − 2 ) = ln ⁡ 2 + i π. How does this work?I'm messing with exponential growth functions. I noticed that I can write y ( t ) = y ( 0 ) α t as y ( t ) = y ( 0 ) e ln ⁡ ( α ) t (and then I can go ahead and replace ln ⁡ ( α ) with λ.)But how do I handle when the α in ln ⁡ ( α ) is negative? WolframAlpha simply evaluates e ln ⁡ ( − 2 ) as − 2but how do I get avoid the whole negative number issue?WolframAlpha also calls log the natural logarithm, which is confusing. :/

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e          ln      ⁡      (      −      2      )        =  −  2 but   ln  ⁡  (  −  2  )  =  ln  ⁡  2  +  i  π. How does this work?I&#039;m messing with exponential growth functions. I noticed that I can write  y  (  t  )  =  y  (  0  )      α    t   as  y  (  t  )  =  y  (  0  )      e          ln      ⁡      (      α      )      t      (and then I can go ahead and replace   ln  ⁡  (  α  ) with   λ.)But how do I handle when the   α in   ln  ⁡  (  α  ) is negative? WolframAlpha simply evaluates      e          ln      ⁡      (      −      2      )       as   −  2but how do I get avoid the whole negative number issue?WolframAlpha also calls log the natural logarithm, which is confusing. :/

Christina Ward · Accepted Answer

You can write any non negative number as an e-power as you did. In case the number is negative, keep the negative outside, so -e^ln(2) The composition of inverse functions is the identity function x, only on their defined domains, so for lnx that means x&amp;gt;0

Hailie Blevins · Accepted Answer

With Euler&#039;s identity,       e          i      θ        =  cos  ⁡  θ  +  sin  ⁡  θ . This is just a nice identity, proven to work for complex numbers. Any complex number can be written then as :where r is its modulus.Now take ln of that:  ln  ⁡  z  =  ln  ⁡  (  r      e          i      θ        )  =  ln  ⁡  r  +  ln  ⁡      e          i      θ        =  ln  ⁡  r  +  i  θNow, because       e          i      θ        =      e          i      (      θ      +      2      k      π      )      , we have in general,  ln  ⁡  z  =  ln  ⁡  r  +  i  (  θ  +  2  k  π  )

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