6. Which of the following functions grow faster than ln(x) as x→∞? Which grow at the same rate as ln(x)? Which grow slower?
a. log_2(x²)

Verified step by step guidance

1. Recall that the natural logarithm function is denoted as \(\ln(x)\) and grows slowly as \(x \to \infty\).
2. Rewrite the given function \(\log_2(x^2)\) in terms of natural logarithms using the change of base formula: \(\log_2(x^2) = \frac{\ln(x^2)}{\ln(2)}\).
3. Simplify \(\ln(x^2)\) using logarithm properties: \(\ln(x^2) = 2 \ln(x)\), so \(\log_2(x^2) = \frac{2 \ln(x)}{\ln(2)}\).
4. Since \(\frac{2}{\ln(2)}\) is a positive constant, \(\log_2(x^2)\) is essentially a constant multiple of \(\ln(x)\), meaning it grows at the same rate as \(\ln(x)\) as \(x \to \infty\).
5. Therefore, \(\log_2(x^2)\) neither grows faster nor slower than \(\ln(x)\); it grows at the same rate.

Key Concepts

Growth Rates of Functions

Understanding how functions behave as their input approaches infinity is essential. Growth rates compare how quickly functions increase, allowing classification into faster, slower, or equivalent growth. For example, polynomial functions grow faster than logarithmic functions as x→∞.

Properties of Logarithms

Logarithmic properties, such as log_b(x^k) = k·log_b(x), help simplify and compare functions. Recognizing that log_2(x²) = 2·log_2(x) shows it differs from ln(x) by a constant multiple, which affects growth rate comparisons.

Asymptotic Equivalence

Two functions grow at the same rate asymptotically if their ratio approaches a nonzero constant as x→∞. This concept helps determine if functions like log_2(x²) and ln(x) have equivalent growth by analyzing their limits and constant factors.