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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 7.3.77c
Chapter 7, Problem 7.3.77c

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)

Verified step by step guidance
1
Recall the logarithm property that states \(\ln(a) = -\ln\left(\frac{1}{a}\right)\), which means \(\ln(a) = -\ln(b)\) if and only if \(b = \frac{1}{a}\).
Identify the two expressions inside the logarithms: \(a = 1 + \sqrt{2}\) and \(b = -1 + \sqrt{2}\).
Check if \(b\) is the reciprocal of \(a\) by calculating \(\frac{1}{a} = \frac{1}{1 + \sqrt{2}}\) and compare it to \(b\).
Rationalize the denominator of \(\frac{1}{1 + \sqrt{2}}\) by multiplying numerator and denominator by the conjugate \(1 - \sqrt{2}\) to simplify the expression.
Compare the simplified form of \(\frac{1}{1 + \sqrt{2}}\) with \(-1 + \sqrt{2}\) to determine if they are equal, which will confirm whether the original statement is true or false.

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Key Concepts

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

Properties of the Natural Logarithm Function

The natural logarithm function ln(x) is defined only for positive real numbers x > 0. It is the inverse of the exponential function e^x, and ln(a) is undefined for a ≤ 0 in the real number system. Understanding its domain is crucial when evaluating expressions involving ln.
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Logarithm Identity: ln(a) = -ln(1/a)

A key logarithmic identity states that ln(a) = -ln(1/a) for positive a. This means that the negative of the logarithm of a number equals the logarithm of its reciprocal. This identity helps in rewriting and comparing logarithmic expressions.
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Evaluating Expressions Involving Square Roots and Logarithms

When dealing with expressions like ln(1 + √2) and ln(-1 + √2), it is important to evaluate the numerical values inside the logarithm to ensure they are positive. Since √2 ≈ 1.414, 1 + √2 > 0 but -1 + √2 ≈ 0.414 > 0, so both arguments are positive, allowing the logarithms to be defined and compared.
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Related Practice
Textbook Question

Chemotherapy In an experimental study at Dartmouth College, mice with tumors were treated with the chemotherapeutic drug Cisplatin. Before treatment, the tumors consisted entirely of clonogenic cells that divide rapidly, causing the tumors to double in size every 2.9 days. Immediately after treatment, 99% of the cells in the tumor became quiescent cells which do not divide and lose 50% of their volume every 5.7 days. For a particular mouse, assume the tumor size is 0.5 cm³ at the time of treatment.

d. Plot a graph of V(t) for 0 ≤ t ≤ 15. What happens to the size of the tumor, assuming there are no follow-up treatments with Cisplatin?

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

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

Terminal velocity Refer to Exercises 95 and 96.


d. How tall must a cliff be so that the BASE jumper (m = 75 kg and k = 0.2) reaches 95% of terminal velocity? Assume the jumper needs at least 300 m at the end of free fall to deploy the chute and land safely.

Textbook Question

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.

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