Logarithm of a transcendental number Can anything be said about the nature of the number log ⁡ y where y is a transcendental number not of the form y = e x or written trivially in that form using x = log ⁡ w for some w transcendental? Would the result always be transcendental? Just considering real numbers here.

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Logarithm of a transcendental number Can anything be said about the nature of the number   log  ⁡  y where   y is a transcendental number not of the form   y  =      e    x   or written trivially in that form using   x  =  log  ⁡  w for some   w transcendental? Would the result always be transcendental? Just considering real numbers here.

Alannah Hanson · Accepted Answer

What can be said about   ln  ⁡  y where   y is not of the form   y  =      e    x  ?That such a value can never exist. Because only negative numbers aren&#039;t of the form   y  =      e    x   and the natural logs of negative numbers don&#039;t exist.Is it always transcendental?No, it is never transcendental because it never exists.If   y is non-negative and it&#039;s not of the form   y  =      e    x    ;  x not transcendental? Is that always transcendental?Yes. If   y is not of the form   y  =      e    x    ;  x not transcendental then   y is of the form   y  =      e    x    ;  x is transcendental, so   ln  ⁡  y  =  x is transcendental.

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