Domanda del libro di testo
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
2. b. tan^(-1)(√3)
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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 7.2.1b
1. Express the following logarithms in terms of ln 2 and ln 3.
b. ln(4/9)
Guida verificata passo dopo passo1
Recall the logarithm property that allows you to express the logarithm of a quotient as the difference of logarithms: \(\ln\left(\frac{a}{b}\right) = \ln a - \ln b\).
Apply this property to the given expression: \(\ln\left(\frac{4}{9}\right) = \ln 4 - \ln 9\).
Express 4 and 9 in terms of their prime factors: \(4 = 2^2\) and \(9 = 3^2\).
Use the logarithm power rule, which states \(\ln(a^b) = b \ln a\), to rewrite \(\ln 4\) and \(\ln 9\) as \(2 \ln 2\) and \(2 \ln 3\) respectively.
Combine these results to express the original logarithm entirely in terms of \(\ln 2\) and \(\ln 3\): \(\ln\left(\frac{4}{9}\right) = 2 \ln 2 - 2 \ln 3\).
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Properties of Logarithms
Logarithms have key properties such as the product, quotient, and power rules. The quotient rule states that ln(a/b) = ln(a) - ln(b), allowing the expression of logarithms of fractions as differences of logarithms.
Video consigliato:
05:36
Change of Base Property
Natural Logarithm (ln)
The natural logarithm, denoted ln, is the logarithm with base e. It is commonly used in calculus and can be manipulated using its properties to simplify expressions involving constants like 2 and 3.
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05:18Derivative of the Natural Logarithmic Function
Expressing Numbers as Powers or Products
To rewrite logarithms in terms of ln 2 and ln 3, numbers like 4 and 9 should be expressed as powers or products of 2 and 3, e.g., 4 = 2^2 and 9 = 3^2, enabling the use of the power rule for logarithms.
Video consigliato:
04:21The Product Rule Example 1
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