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2. Which of the following functions grow faster than e^x as x→∞? Which grow at the same rate as e^x? Which grow slower?
e. e^(-x)
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Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 7, Problema 7.2.2d
2. Express the following logarithms in terms of ln 5 and ln 7.
d. ln 1225
Guida verificata passo dopo passo1
Start by factoring the number inside the logarithm to express it in terms of its prime factors or known bases. For 1225, find its prime factorization.
Recognize that 1225 can be written as a product of powers of 5 and 7, since 1225 = 35^2 and 35 = 5 \(\times\) 7.
Rewrite the logarithm using the property of logarithms that states \( \ln(a^b) = b \ln(a) \). So, \( \ln(1225) = \ln(35^2) = 2 \ln(35) \).
Next, use the logarithm product rule \( \ln(ab) = \ln(a) + \ln(b) \) to express \( \ln(35) \) as \( \ln(5) + \ln(7) \).
Combine all the steps to express \( \ln(1225) \) entirely in terms of \( \ln(5) \) and \( \ln(7) \) as \( 2 (\ln(5) + \ln(7)) \).
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Properties of Logarithms
Logarithmic properties such as the product, quotient, and power rules allow us to break down complex logarithms into simpler parts. For example, ln(ab) = ln a + ln b and ln(a^n) = n ln a. These rules help express logarithms in terms of known values.
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05:36
Change of Base Property
Prime Factorization
Prime factorization involves expressing a number as a product of prime numbers. This is useful in logarithms to rewrite the argument as a product of primes or their powers, enabling the use of logarithmic properties to simplify expressions.
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Percorso guidato
11:26Partial Fraction Decomposition: Repeated Linear Factors
Natural Logarithm (ln)
The natural logarithm, denoted ln, is the logarithm with base e. Understanding that ln is a specific logarithm function helps in manipulating and expressing logarithms in terms of known ln values, such as ln 5 and ln 7.
Video consigliato:
05:18Derivative of the Natural Logarithmic Function
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In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:
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In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:
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72. y= 2-x-x³, -2 ≤ x ≤ 2, x_0 = 3/2
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Domanda del libro di testo
In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:
d. Find the equation for the tangent line to g at the point (f(x_0), x_0) located symmetrically across the 45° line y=x (which is the graph of the identity function). Use Theorem 1 to find the slope of this tangent line.
67. y= √(3x-2), 2/3 ≤ x ≤ 4, x_0=3
25
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Domanda del libro di testo
In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:
d. Find the equation for the tangent line to g at the point (f(x_0), x_0) located symmetrically across the 45° line y=x (which is the graph of the identity function). Use Theorem 1 to find the slope of this tangent line.
70. y= x³/(x²+1), -1 ≤ x ≤ 1, x_0=1/2
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Domanda del libro di testo
6. Which of the following functions grow faster than ln(x) as x→∞? Which grow at the same rate as ln(x)? Which grow slower?
e. x - 2ln(x)
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