e^pi or pi^e: How to determine which is bigger without a calculator?

apple2357

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Does anyone have any ideas how i could find whether e^pi is bigger than pi^e, without using a calculator?
I am thinking about a function perhaps where i might be able to substitute something in and compare , but i haven't got anywhere with it?
Any creative ideas out there?
 
Does anyone have any ideas how i could find whether e^pi is bigger than pi^e, without using a calculator?
I am thinking about a function perhaps where i might be able to substitute something in and compare , but i haven't got anywhere with it?
Any creative ideas out there?

Have someone else use a calculator and tell you?

You can add a logarithm transformation and ponder a different problem.

\(\displaystyle \log\left(\pi^{e}\right) = e\cdot\log(\pi)\)

\(\displaystyle \log\left(e^{\pi}\right) = \pi\cdot\log(e) = \pi\)


Anyway, part of it is easier.
 
Have someone else use a calculator and tell you?

You can add a logarithm transformation and ponder a different problem.

\(\displaystyle \log\left(\pi^{e}\right) = e\cdot\log(\pi)\)

\(\displaystyle \log\left(e^{\pi}\right) = \pi\cdot\log(e) = \pi\)


Anyway, part of it is easier.

Ok this looks helpful...thanks
 
I have started playing with y=1/x lnx and thinking about the max point. I think there is something in this too.
It feels simpler than the video.
 
I have started playing with y=1/x lnx and thinking about the max point. I think there is something in this too.
It feels simpler than the video.
It is great that you are playing with different functions or at least one function. Why do you think this function will help? Please let us know.


Most importantly is your function y = 1/(xlnx) or y= (1/x)*lnx????????
 
It is great that you are playing with different functions or at least one function. Why do you think this function will help? Please let us know.


Most importantly is your function y = 1/(xlnx) or y= (1/x)*lnx????????

It is y= (1/x)*lnx

I looked at the graph and noticed it has a maximum value at x=e .. haven't seen anything else yet.
The inspiration came from the video , which looks at y=x^x, which feels more sophisticated to me.
 
It is y= (1/x)*lnx

I looked at the graph and noticed it has a maximum value at x=e .. haven't seen anything else yet.
The inspiration came from the video , which looks at y=x^x, which feels more sophisticated to me.

Ok, i think i have cracked this. Can someone check my work?

The max point of y=(1/x)*lnx is at (e, 1/e*lne) - ( product rule differentiation )

And when x=pi on y= (1/x)*lnx we have the y coordinate (1/pi)*ln(pi)

But since y= 1/e*lne is the max value of the function

we can say
(1/e)* lne > (1/pi)*ln pi

pi* lne > e* ln pi

ln e^pi > ln pi^e


so e^pi> pi^e


Does this seem ok?
 
Ok, i think i have cracked this. Can someone check my work?

The max point of y=(1/x)*lnx is at (e, 1/e*lne) - ( product rule differentiation )

And when x=pi on y= (1/x)*lnx we have the y coordinate (1/pi)*ln(pi)

But since y= 1/e*lne is the max value of the function

we can say
(1/e)* lne > (1/pi)*ln pi

pi* lne > e* ln pi

ln e^pi > ln pi^e


so e^pi> pi^e


Does this seem ok?
Nicely done.
 
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