- Difficulty
- Standard
- Suggested time
- 4 min
- Marks
- 2
Power functions and first sketches practice question 11
Each of , and grows without bound as . For all sufficiently large , which ordering is correct, from largest to smallest?
Each of the three belongs to a family: exponentials, powers of , or powers of . How do those families rank in the long run, however close the base is to 1 or however small the exponent?
Write . Is that a growing exponential, and how does a growing exponential compare with any power of ?
Any positive power of eventually beats , even . Raise both sides of to the fourth power.
Theory UBC Math 100 §1.5Key results
Worked solution
Strategy
Use the long-run pecking order: a growing exponential beats every power, and every positive power beats the logarithm, even when the logarithm is raised to a power. Small bases and small exponents only delay the overtaking.
Working
A base above 1, however close to 1, gives a growing exponential.
Every growing exponential eventually overtakes every power.
Any positive power eventually beats log x; raising to the fourth power keeps the order because both sides are positive.
The overtaking happens late, around x = e26, about 2 × 10¹¹, which is why testing one number misleads; the question is about the long run.
Answer
The answer is 1.005 to the x, then the square root of x, then the fourth power of natural log of x.
Check
At x = 1012, (log x)⁴ ≈ 583,000 is below √x = 1,000,000, and 1.005x has over two billion digits.
Where this goes wrong
Ranking by trying one large number. At x = 10⁶ the fourth power of the logarithm is far above √x, but the long-run order is fixed: growing exponentials beat powers, and positive powers beat any power of log x.
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