Egative power indicates an inverse

[gillgill]

Why is 3^x > e^x > 2^x when x>0, but 3^x < e^x < 2^x when x<0?

 

[moose]

e is approx. 2.72

so 3^1=3
2.72^1=2.72
2^1=2

therefore 3^x>e^x>2^x when x>0

however, when you get into the negatives

3^-1=1/3
2.72^-1=1/2.72
2^-1=1/2


this is because a negative power indicates an inverse( 1/x ), so the smaller the number with a negative power, the smaller the denomenator will be

 

[EnumaElish]

Why is 3^x > e^x > 2^x when x>0, but 3^x < e^x < 2^x when x<0?

Another way of looking at it. Since 3 > e and therefore Log(3) > 1, x Log(3) > x implies x > 0 and x Log(3) < x implies x < 0.

 

[HallsofIvy]

All this is saying is that if 0<a< b< c then 0< 1/c< 1/b< 1/a

If a< b and a and b are positive, then 1< b/a because we have divided by a positive number.

Then 1/b< 1/a because we have divided by a positive number.