5. 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_3(x)

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Recall that logarithms with different bases differ only by a constant multiple. Specifically, for any base \(a > 0, a \neq 1\), \(\log_a(x) = \frac{\ln(x)}{\ln(a)}\).

Since \(\log_3(x) = \frac{\ln(x)}{\ln(3)}\), it is exactly \(\ln(x)\) multiplied by a positive constant \(\frac{1}{\ln(3)}\).

When comparing growth rates as \(x \to \infty\), multiplying by a positive constant does not change the rate of growth, so \(\log_3(x)\) and \(\ln(x)\) grow at the same rate.

Therefore, \(\log_3(x)\) neither grows faster nor slower than \(\ln(x)\); they grow at the same rate asymptotically.

In summary, \(\log_3(x)\) grows at the same rate as \(\ln(x)\) as \(x \to \infty\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Growth Rates of Logarithmic Functions

Logarithmic functions with different bases differ only by a constant factor. For example, log base 3 of x can be expressed as ln(x) divided by ln(3). This means all logarithmic functions grow at the same rate asymptotically, differing only by a constant multiplier.

Asymptotic Behavior and Limits

As x approaches infinity, comparing growth rates involves analyzing limits of ratios of functions. If the limit of f(x)/g(x) is a positive finite constant, f and g grow at the same rate. If the limit is zero or infinity, one grows slower or faster respectively.

Big-O and Little-o Notation

Big-O and little-o notation describe relative growth rates of functions. If f(x) = O(g(x)), f grows no faster than g asymptotically. If f(x) = o(g(x)), f grows strictly slower than g. These notations help classify functions like logarithms, polynomials, and exponentials by their growth.

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