proof of Irrationality of e

Mar 2015
2
0
Lake Placid
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.
 

skeeter

Math Team
Jul 2011
3,130
1,697
Texas
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?
 
Mar 2015
2
0
Lake Placid
Thanks a lot. I misused the change base formula