Domanda del libro di testo
Evaluate the integrals in Exercises 67–74 in terms of
b. natural logarithms.
69. ∫(from 5/4 to 2)dx/(1-x²)
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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 7.2.2b
2. Express the following logarithms in terms of ln 5 and ln 7.
b. ln 9.8
Guida verificata passo dopo passo1
Start by expressing 9.8 as a product or quotient of numbers involving 5 and 7. Notice that 9.8 can be written as \( 9.8 = \frac{49}{5} \) because \( 49 = 7^2 \) and \( 9.8 = 9.8 \).
Rewrite \( \ln 9.8 \) using the expression found: \( \ln 9.8 = \ln \left( \frac{49}{5} \right) \).
Use the logarithm property for division: \( \ln \left( \frac{a}{b} \right) = \ln a - \ln b \). So, \( \ln 9.8 = \ln 49 - \ln 5 \).
Express \( \ln 49 \) in terms of \( \ln 7 \) using the power rule for logarithms: \( \ln 49 = \ln (7^2) = 2 \ln 7 \).
Combine all parts to write \( \ln 9.8 \) fully in terms of \( \ln 5 \) and \( \ln 7 \): \( \ln 9.8 = 2 \ln 7 - \ln 5 \).
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Properties of Logarithms
Logarithmic properties such as the product, quotient, and power rules allow us to rewrite logarithms of complex numbers in terms of simpler components. For example, ln(ab) = ln a + ln b and ln(a^n) = n ln a. These properties are essential for expressing ln 9.8 in terms of ln 5 and ln 7.
Video consigliato:
05:36
Change of Base Property
Prime Factorization and Approximation
Breaking down numbers into products of prime factors or close approximations helps in rewriting logarithms. Since 9.8 is not a simple product of 5 and 7, approximating or expressing 9.8 as a product or quotient involving 5 and 7 is necessary to use known logarithms effectively.
Video consigliato:
Percorso guidato
11:15Partial Fraction Decomposition: Repeated Linear Factors Example 4
Natural Logarithm (ln) Definition
The natural logarithm ln x is the logarithm to the base e, where e is Euler's number (~2.718). Understanding ln as the inverse of the exponential function helps in manipulating and combining logarithmic expressions, especially when expressing one logarithm in terms of others.
Video consigliato:
05:18Derivative of the Natural Logarithmic Function
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