π = Pi = 22/7 (22/7 is approximate value of pi more accurate value is 355/113)
Now we will take mathematical approach in solving the equality problem.
Lets Assume pi^e > e^pi
now taking natural log (Log of base e) on both sides.
Log (pi^e) > Log (e^pi)
eLog(pi) > piLog(e)
eLog(pi) > pi ---> Log e is 1 because its Log of base e. (LOGe(e) = 1)
Log(pi) > pi/e
since we know that Pi > 1 and hence Log(pi) >1,
then above becomes
1 > pi/e
e > pi - which is false (pi = 3.14, e = 2.72)
Therefore our assumption (pi^e > e^pi) is not correct
hence pi^e is less than e^pi.
This is incorrect on two counts.
ReplyDelete1. The statement "since we know that Pi > 1 and hence Log(pi) >1" is wrong. All we can say from this is that pi > 1 means log(pi) > 0. We could say instead that pi > e thus log(pi) > log(e) = 1
2. However, log(pi) > 1 doesn't help since this inequality *does not* imply 1 > pi/e. Why, since we don't know the value of log(pi) a priori it could be, say, 1.001, which is certainly not greater than pi/e.
The above "solution" only works if we use a calculator (or other computational means) to determine the result numerically. If I want to do that, I will simply use a calculator directly.
Here is a way to do the problem without using calculus:
ReplyDeleteexp(0) = 1 => exp(1) = e > 1 (1)
pi > 0 => ln(pi) < pi (2)
From (1) and (2):
ln(pi) < pi*1 < pi*e
pi < e^(pi*e) [e^(pi*e) = (e^e)^pi]
pi^(1/pi) < e^e
ln(pi^(1/pi)) < ln(e^e) [ln(e^e) = e*ln(e) = e]
ln(ln(pi^(1/pi))) < ln(e) [ln(e) = 1]
ln(ln(pi)/pi)) < 1
ln(ln(pi)) - ln(pi) < 1
ln(pi) - ln(ln(pi)) > 1 [i.e., multiply each side of the
line above by -1]
ln(pi/ln(pi)) > 1
pi/ln(pi) > e
pi/e > ln(pi)
pi > e*ln(pi)
pi > ln(pi^e)
e^pi > pi^e
Actually, the above solution I gave is wrong also, as the line ln(pi) – ln(ln(pi)) > 1 should have -1 on the right side, so it should not be posted.
ReplyDelete