Which growth order correctly ranks exponential, polynomial, and logarithmic functions as x grows large?

Which growth order correctly ranks exponential, polynomial, and logarithmic functions as x grows large?

• A. Exponential, Polynomials, Logs
• B. Logs, Exponential, Polynomials
• C. Exponential, Logs, Polynomials
• D. Polynomials, Exponential, Logs

Answer: (C)

When x gets really large, compare how fast each function goes to infinity. Exponential growth dominates every polynomial: for any base a > 1 and any positive n, the ratio a^x / x^n tends to infinity as x → ∞. So exponentials outrun polynomials. Next, polynomials outrun logarithms: for any n > 0, x^n / log x → ∞ as x → ∞. Therefore, the fastest to slowest among these is exponential, then polynomial, then logarithmic growth. The standard ranking from fastest to slowest is exponential, polynomial, logarithmic.

When x gets really large, compare how fast each function goes to infinity. Exponential growth dominates every polynomial: for any base a > 1 and any positive n, the ratio a^x / x^n tends to infinity as x → ∞. So exponentials outrun polynomials. Next, polynomials outrun logarithms: for any n > 0, x^n / log x → ∞ as x → ∞. Therefore, the fastest to slowest among these is exponential, then polynomial, then logarithmic growth. The standard ranking from fastest to slowest is exponential, polynomial, logarithmic.