ln x is unbounded Use the following argument to show that lim (x → ∞) ln x = ∞ and lim (x → 0⁺) ln x = −∞.
c. Show that ln 2ⁿ > n/2 and ln 2^(−n) < −n/2.

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Recall the properties of logarithms, specifically that \( \ln(a^b) = b \ln a \). Use this to rewrite \( \ln 2^n \) as \( n \ln 2 \) and \( \ln 2^{-n} \) as \( -n \ln 2 \).

Note that \( \ln 2 \) is a positive constant since 2 is greater than 1. To show \( \ln 2^n > \frac{n}{2} \), compare \( n \ln 2 \) with \( \frac{n}{2} \). This reduces to showing \( \ln 2 > \frac{1}{2} \).

Similarly, to show \( \ln 2^{-n} < -\frac{n}{2} \), compare \( -n \ln 2 \) with \( -\frac{n}{2} \). This reduces to showing \( \ln 2 > \frac{1}{2} \) again, which is consistent with the previous step.

Since \( \ln 2 \) is approximately 0.693, which is greater than 0.5, both inequalities hold true for all positive integers \( n \).

These inequalities help illustrate the behavior of \( \ln x \) as \( x \to \infty \) and \( x \to 0^+ \) by bounding the logarithm with linear functions that tend to infinity or negative infinity, respectively.

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Behavior of the Natural Logarithm Function

The natural logarithm function, ln(x), increases without bound as x approaches infinity, and decreases without bound as x approaches zero from the right. This means lim (x → ∞) ln(x) = ∞ and lim (x → 0⁺) ln(x) = −∞, reflecting its unbounded growth and decay.

Properties of Exponents and Logarithms

Logarithms and exponents are inverse operations. For any positive base a, ln(a^n) = n ln(a). This property allows us to rewrite expressions like ln(2^n) as n ln(2), facilitating comparisons and inequalities involving exponential and logarithmic terms.

Inequalities Involving Logarithms

To show inequalities such as ln(2^n) > n/2, one uses known bounds or approximations of ln(2). Since ln(2) ≈ 0.693 > 0.5, multiplying by n preserves the inequality. Similarly, for negative exponents, ln(2^{-n}) = -n ln(2) < -n/2, demonstrating how logarithmic inequalities relate to linear bounds.

ln x is unbounded Use the following argument to show that lim (x → ∞) ln x = ∞ and lim (x → 0⁺) ln x = −∞.c. Show that ln 2ⁿ > n/2 and ln 2^(−n) < −n/2.