Which one is bigger?

Question

℮^π or π^℮. Plz give a mathematical explanation for your answer.

'e' is a mathematical constant which is used as the base of the natural logarithm.
The value of 'e' lies between 2 and 3.So,
e < π.

Answer 1

let us take x^(1/x)
let y = x^(1/x)
ln y = ln x/x
x ln y = ln x
differentiate both sides
x/y dy/dx + ln y = 1/x
dy/dx = (1/x - ln y)/(x/y)
= (1/x-1/x ln x)/(x/y) = y/x^2(1-ln x)
for x > e this is -ve
so x^(1/x) decreases

so e^(1/e) > pi^(1/pi)
raise both to the power e * pi

e^pi > pi^ e

Answer 2

Use scientific calculator or computer you will get e^pi > pi^e.
Numerical computation is the best mathematical explaination for any mathematical problem (if possible).

Answer 3

Hi:

I am by no means a math whiz but I have given it a shot.

Here goes...

The answer depends on the value of 'e'.

If 'e' is a positive whole number equal to or greater than 1, or when it is a decimal or a fraction , like 0.5, then pi^e is greater.

If it is a negative number, then the first expression is invalid.

Hope this answers your question.

Cheers!!

Hey, Jaques is right... just remembered that e is a constant

Answer 4

e ^pi is bigger.

Let's prove it.
e ^pi > pi ^e iff
pi log e > e log pi
iff
pi/log pi > e.
Look at the function f(x) = x/log(x).
By looking at its first derivative
(log x -1)/(log(x)^2), we see
that the function is decreasing for 0 < x < e,
has a minimum value at x = e and is increasing
for x >e. Since pi > e, pi/log pi > e,
so e^pi > pi ^e. (No calculator needed!)

Answer 5

try to apply logarithm to the base e to both .then u will find that the log of e^pi is pi but for pi^e the value is 1.1447, which is less than pi hence e^pi is greater than pi^e.

Answer 6

consider f(x) = ln (e^x/x^e)
= x - e*lnx
f'(x)= 1 -(e/x)
f'(x) = 0 implies x = e
f''(x) is - e/(x^2)
which is less than 0 for x = e
this implies max value of fn. is f(e) which is zero
so for any other x, f(x) will be less than zero
this means f(pi)<0

ln (e^pi/pi^e) < 0
and that means e^pi/pi^e <1

hence e^pi < pi^e

kp_math just chk ur differentiation....

Answer 7

actually..therre is no meaning for the function e^pi....pi^e...they both are the syntax errors..
hence there values cannot be defined...
the value of pi=3.141592654 approx
and
e=2.71828

the gentleman who answererd above me is just a crap...
how can the value of an exponential function be negitive...
the min. value of all exponential functions is 0 at x=-infinity

Answer 8

e^pi = ~23.140693
pi^e = ~22.459158

to the other commenters, e is a mathematical constant
http://en.wikipedia.org/wiki/E_%28mathematical_constant%29

Answer 9

Pie^e cause e is a variable that stands for any number so if you raised pie by a google then it would be a very big number but if you (google^pie) then it is a number almost 456.4654 as small

Answer 10

Assuming e is your variable then as long as e>pi pi^e would be larger.