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)

Guida verificata passo dopo passo

Recall that the natural logarithm function is denoted as \(\ln(x)\) and grows slowly as \(x \to \infty\).

Rewrite the given function \(\log_2(x^2)\) in terms of natural logarithms using the change of base formula: \(\log_2(x^2) = \frac{\ln(x^2)}{\ln(2)}\).

Simplify \(\ln(x^2)\) using logarithm properties: \(\ln(x^2) = 2 \ln(x)\), so \(\log_2(x^2) = \frac{2 \ln(x)}{\ln(2)}\).

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 \(x \to \infty\).

Therefore, \(\log_2(x^2)\) neither grows faster nor slower than \(\ln(x)\); it grows at the same rate.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Durata del video:

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

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→∞.

Video consigliato:

Percorso guidato

04:16

Intro To Related Rates

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.

Video consigliato:

05:36

Change of Base Property

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.

Video consigliato:

Percorso guidato