Domanda del libro di testo
3. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?
g. x^3 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 7.2.2f
2. Express the following logarithms in terms of ln 5 and ln 7.
f. (ln35 + ln(1/7))/(ln25)
Guida verificata passo dopo passo1
Recall the logarithm properties: \(\ln(ab) = \ln a + \ln b\) and \(\ln\left(\frac{a}{b}\right) = \ln a - \ln b\).
Rewrite \(\ln 35\) as \(\ln(5 \times 7)\), which becomes \(\ln 5 + \ln 7\) using the product rule.
Rewrite \(\ln \left(\frac{1}{7}\right)\) as \(\ln 1 - \ln 7\). Since \(\ln 1 = 0\), this simplifies to \(-\ln 7\).
Add the terms in the numerator: \((\ln 35 + \ln(1/7)) = (\ln 5 + \ln 7) + (-\ln 7) = \ln 5\).
Rewrite the denominator \(\ln 25\) as \(\ln(5^2) = 2 \ln 5\). Then express the entire fraction as \(\frac{\ln 5}{2 \ln 5}\).
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Properties of Logarithms
Logarithms have key properties such as the product rule (ln a + ln b = ln(ab)), the quotient rule (ln a - ln b = ln(a/b)), and the power rule (ln(a^b) = b ln a). These allow complex logarithmic expressions to be simplified or rewritten in terms of simpler logarithms.
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05:36
Change of Base Property
Change of Base and Logarithm Simplification
Simplifying logarithmic expressions often involves rewriting numbers as products or powers of known bases. For example, expressing 35 as 5 × 7 or 25 as 5² helps rewrite logarithms in terms of ln 5 and ln 7, facilitating easier manipulation and evaluation.
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05:36Change of Base Property
Natural Logarithm (ln) and Its Use
The natural logarithm, denoted ln, is the logarithm with base e. It is commonly used in calculus and higher mathematics. Understanding how to express other logarithms in terms of ln 5 and ln 7 requires familiarity with ln’s properties and how to decompose numbers into factors involving 5 and 7.
Video consigliato:
05:18Derivative of the Natural Logarithmic Function
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