October 27th, 2018, 12:09 PM  #1 
Senior Member Joined: Nov 2011 Posts: 247 Thanks: 3  e^n
Is the next sentence true or false: Every e^(number) equations are transcendental equation. 
October 27th, 2018, 12:19 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,264 Thanks: 1198 
I wouldn't say that $e^0 = 1$ is transcendental.

October 27th, 2018, 12:32 PM  #3 
Senior Member Joined: Nov 2011 Posts: 247 Thanks: 3  
October 27th, 2018, 12:33 PM  #4 
Senior Member Joined: Oct 2009 Posts: 688 Thanks: 223  
October 27th, 2018, 12:47 PM  #5  
Senior Member Joined: Nov 2011 Posts: 247 Thanks: 3  Quote:
I ask on other thing. The first question in about number that I don't understand if there is counterexample to the sentence... And in second question I ask about particular thing, the complex number. The second question is about all the complex number and my question is about counterexample of it. So why question is: Why it can't be a counterexample in the second question? Is There a reason to it?  
October 27th, 2018, 01:02 PM  #6 
Senior Member Joined: Sep 2015 From: USA Posts: 2,264 Thanks: 1198 
$e^{\ln(k)} = k,~\forall k \in \mathbb{Z},~k>0$ Do you consider this transcendental? Certainly $k$ isn't. $\ln(k)$ however probably is 
October 27th, 2018, 01:43 PM  #7 
Senior Member Joined: Nov 2011 Posts: 247 Thanks: 3  
October 27th, 2018, 01:45 PM  #8 
Senior Member Joined: Oct 2009 Posts: 688 Thanks: 223  
October 27th, 2018, 02:02 PM  #9 
Senior Member Joined: Sep 2015 From: USA Posts: 2,264 Thanks: 1198  
October 27th, 2018, 05:12 PM  #10  
Senior Member Joined: Aug 2012 Posts: 2,134 Thanks: 621  Quote:
First, if $n$ is a number  integer, real, or complex  then $e^n$ is another number. It's not an equation. So it might or might not be a transcendental number, but it's always one single, particular number. Not an equation. We could ask when $e^n$ is transcendental. It's pretty clear that if $n$ is an integer, $e^n$ is transcendental if and only if $n$ is nonzero. If $n$ is real, $e^n$ is transcendental unless we specifically select $n$ so it's not. For example if $n$ = $\ln 2$, then $e^n = 2$, which is not transcendental. Of course I haven't given or thought about the exact criteria by which we can know if a real exponent leads to a transcendental number. But you can always use a natural log to undo raising $e$ to a power. And vice versa. That's the most important thing to know about log and exp. They undo each other. If $n$ is complex it's much the same. Some specifically cookup values will give you nontranscendentals (aka algebraics) and "random" complex numbers will generally lead to transcendental complex numbers. So that's one problem I have with your question. The way you phrased it, it refers to numbers and not equations. Secondly, what is a transcendental equation? Do you mean it in the sense of https://en.wikipedia.org/wiki/Transcendental_equation? Or do you mean $e^z$ as a function? So your question is ambiguous in a couple of ways and that's perhaps why you are not happy with the responses you're getting. Last edited by Maschke; October 27th, 2018 at 05:14 PM.  