Power functions and first sketches practice question 10
For all sufficiently large , which ordering of the three functions , and is correct, from largest to smallest?
The question is about eventual behaviour, not about any particular x. Which family grows fastest in the long run: logarithms, powers or exponentials?
Compare with by looking at how each changes when goes up by 1: one is multiplied by , the other by , which tends to 1.
Theory UBC Math 100 §1.5Key results
Worked solution
Strategy
The three families have a fixed pecking order for large x: exponentials beat every power, and every positive power beats the logarithm. Coefficients and exponents change where the overtaking happens, not whether it happens.
Working
Each step of 1 in x multiplies ex by a fixed factor above 1, while the factor for x¹00 shrinks toward 1. Eventually the exponential wins.
The logarithm grows more slowly than any positive power; it is below x, and x is below x¹00 once x > 1.
The exponential overtakes x¹00 only for very large x (around x = 650), but the question is about large x.
Answer
The answer is e to the x, then x to the 100, then natural log of x.
Check
At x = 1000, e1000 is more than 10434, far above x100 = 10300, while log x is under 7.
Where this goes wrong
Testing x = 100 and finding x¹00 = 10²00 far above e¹00 ≈ 2.7 × 104³. That is true at x = 100, but the question asks about all sufficiently large x, and the exponential takes over eventually, whatever the power.
The idea behind it
The power functions and first sketches lesson, a figure to move, in a few steps
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Let . For very large positive , which term dominates, and what is the sign of ?Saved in this browser only.