已知2^x=3^y=5^z,且x,y,z都是正数,试比较2x,3y,5z的大小

Question

已知2^x=3^y=5^z,且x,y,z都是正数,试比较2x,3y,5z的大小

Answer 1

2^x = 3^y = 5^z = a > 1
x = log_2(a), y = log_3(a), z = log_5(a)
2x = 2log_2(a) = 2*ln(a)/ln2
3y = 3log_3(a) = 3*ln(a)/ln3
5z = 5log_5(a) = 5*ln(a)/ln5
2*ln(5) = ln25, 5*ln(2) = ln(32)
2*ln(5) < 5ln(2), 2/ln(2) < 5/ln(5), ln(a) > 0
2x < 5z (1)
3*ln(5) = ln(125), 5*ln(3) = ln(243)
3*ln(5) < 5*ln(3)
3y < 5z (2)
2*ln(3) = ln(9), 3*ln(2) = ln(8)
2*ln(3)> 3*ln(2)
2x > 3y (3)
3y < 2x < 5z