Skip to main content
Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblema 7.3.77c
Capitolo 7, Problema 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)

Guida verificata passo dopo passo
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.

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:
2m

Concetti chiave

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

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.
Video consigliato:
Percorso guidato
06:21
Properties of Functions

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.
Video consigliato:
7:17
Verifying Trig Equations as Identities

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.
Video consigliato:
5:14
Evaluate Logarithms
Pratica correlata
Domanda del libro di testo

Acceleration, velocity, position Suppose the acceleration of an object moving along a line is given by a(t) = -k v(t), where k is a positive constant and v is the object's velocity. Assume the initial velocity and position are given by v(0) = 10 and s(0) = 0, respectively.

c. Use the fact that dv/dt = (dv/ds)(ds/dt) (by the Chain Rule) to find the velocity as a function of position.

108
views
Domanda del libro di testo

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

62
views
Domanda del libro di testo

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

59
views
Domanda del libro di testo

Velocity of falling body Refer to Exercise 95, which gives the position function for a falling body. Use m = 75 kg and k = 0.2.


c. How long does it take for the BASE jumper to reach a speed of 45 m/s (roughly 100 mi/hr)?

39
views
Domanda del libro di testo

Definitions of hyperbolic sine and cosine Complete the following steps to prove that when the x- and y-coordinates of a point on the hyperbola x² - y² = 1 are defined as cosh t and sinh t, respectively, where t is twice the area of the shaded region in the figure, x and y can be expressed as

x = cosh t = (eᵗ + e⁻ᵗ) / 2 and y = sinh t = (eᵗ - e⁻ᵗ) / 2.



b. In Chapter 8, the formula for the integral in part (a) is derived:

∫ √(z² − 1) dz = (z/2)√(z² − 1) − (1/2) ln|z + √(z² − 1)| + C.

Evaluate this integral on the interval [1, x], explain why the absolute value can be dropped, and combine the result with part (a) to show that:

t = ln(x + √(x² − 1)).

96
views
Domanda del libro di testo

Power lines A power line is attached at the same height to two utility poles that are separated by a distance of 100 ft; the power line follows the curve ƒ(x) = a cosh x/a. Use the following steps to find the value of a that produces a sag of 10 ft midway between the poles. Use a coordinate system that places the poles at x = ±50.

c. Use your answer in part (b) to find a, and then compute the length of the power line.

32
views