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March 9th, 2015, 06:27 PM  #1 
Newbie Joined: Mar 2015 From: Lake Placid Posts: 2 Thanks: 0  proof of Irrationality of e
I'm an AP Calc BC student. We just learned Euler's way to prove the irrationality of e. I just suddenly came up with this weird proof in class. But I couldn't find anything wrong with it. Can you guys check this proof and point out any mistake hopefully? Thanks a lot! if e is rational, then e=m/n(simplest form), where m,n are unequal integers(n is not 0) then ln(e)=ln(m/n) then 1=ln(m/n)=logn(m) that turns out m=n, which is not true and also contradicted to "m,n are unequal integers" then e is irrational. 
March 9th, 2015, 06:47 PM  #2 
Math Team Joined: Jul 2011 From: Texas Posts: 2,982 Thanks: 1575 
Are you saying $\displaystyle \ln\left(\frac{m}{n}\right)=\log_n{m}$ ? If so, that isn't true ... $\displaystyle \ln\left(\frac{m}{n}\right) \ne \frac{\ln{m}}{\ln{n}}=\log_n{m}$ $\displaystyle \ln\left(\frac{m}{n}\right) = \ln{m}\ln{n}$, remember? 
March 10th, 2015, 03:47 AM  #3 
Newbie Joined: Mar 2015 From: Lake Placid Posts: 2 Thanks: 0 
Thanks a lot. I misused the change base formula


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