Skip to main content
Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.1.74c
Chapter 7, Problem 7.1.74c

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.

Verified step by step guidance
1
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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1m
Was this helpful?

Key Concepts

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

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.
Recommended video:
05:18
Derivative of the Natural Logarithmic Function

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.
Recommended video:
05:36
Change of Base Property

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.
Recommended video:
07:01
Integrals Involving Natural Logs: Substitution
Related Practice
Textbook Question

Properties of exp(x) Use the inverse relations between ln x and exp(x), and the properties of ln x, to prove the following properties:


c. (exp(x))ᵖ = exp(px), p rational

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

c. ln(1 + √2) = −ln(−1 + √2)

Textbook Question

Oil consumption Starting in 2018 (t=0), the rate at which oil is consumed by a small country increases at a rate of 1.5%/yr, starting with an initial rate of 1.2 million barrels/yr.


c. How many years after 2018 will the amount of oil consumed since 2018 reach 10 million barrels?

Textbook Question

Evaluating hyperbolic functions Use a calculator to evaluate each expression or state that the value does not exist. Report answers accurate to four decimal places to the right of the decimal point.

c. csch⁻¹ 5

Textbook Question

Energy consumption On the first day of the year (t=0), a city uses electricity at a rate of 2000 MW. That rate is projected to increase at a rate of 1.3% per year.


c. Find a function that gives the total energy used (in MW-yr) between t=0 and any future time t>0.

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume x > 0 and y > 0.


c. ln (x + y) = ln x + ln y