e의 π제곱과 π의 e제곱 중 누가 더 클까?

 

중고등학교 과정에서 가장 많이 다루는 무리수는 e와 π가 있다.

e=2.71…인 무리수로 자연상수라고 불리며 일상생활에서의 횟수와 성장률에 대한 정보를 담고 있다.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               

π= 3.14…인 무리수로 원의 지름과 원주사이의 관계를 나타내는 정보를 담고 있다.

 

π>e 임은 너무나 자명하게 알 수 있는데 e의 π제곱과 π의 e제곱을 비교하면 어떨까?

 

막상 누가 더 큰지 대답하려고 한다면 바로 대답하기에는 조금 어려움을 느낄 수 있다.

흔히 대소비교는 빼서 0보다 큰지 작은지 확인하거나 나누어서 1보다 큰지 작은지 비교하거나 해서 확인하지만 이 두 수는 그렇게 비교하기 어렵다. 그렇다면 어떻게 이 두 수의 대소비교를 할 수 있을까?

 

이 두 수를 비교하기 위해서 마지막 방법인 그린다를 사용해보자.

f(x)=lnx/x라는 함수를 가져와보자.

이 함수의 개형은 다음과 같이 생겼으며 f'(x)=0이 되는점을 조사해보면 f'(x)=1-lnx/x^2이므로 x=e에서 최댓값 1/e를 가진다.

자 이때 f에 파이를 대입하면 f(π)=lnπ/π이고 f의 최솟값은 1/e 이므로 1/e > ln π / π이다.

 

이제 이 식을 정리해보자. 양변에 π와 e를 곱하면 식이 이렇게 되고, π > e lnπ

로그의 성질을 이용하면. π > lnπ^e라고 쓸 수 있다.

밑을 e로 해서 로그를 없애면 e^π > π^e라고 할 수 있다.

따라서 e의 π제곱과 π의 e제곱을 비교하면

e의 π제곱이 더 크다는 사실을 알 수 있다.