Domanda del libro di testo
91. [Technology Exercise] 91. The continuous extension of to (sin x)^x to [0, π]
b. Verify your conclusion in part (a) by finding lim(x→0⁺)f(x) with l’Hôpital’s Rule.
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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 7.2.3b
3. Use the properties of logarithms to write the expressions in Exercises 3 and 4 as a single term.
b. ln(3x² - 9x) + ln(1/3x)
Guida verificata passo dopo passo1
Recall the logarithm property that states: \(\ln(a) + \ln(b) = \ln(ab)\). This allows us to combine the sum of logarithms into a single logarithm of the product.
Identify the two expressions inside the logarithms: the first is \(3x^{2} - 9x\) and the second is \(\frac{1}{3}x\).
Multiply the two expressions inside the logarithms: \((3x^{2} - 9x) \times \left(\frac{1}{3}x\right)\).
Simplify the product by distributing and combining like terms: multiply \$3x^{2}$ by \(\frac{1}{3}x\) and $-9x$ by \(\frac{1}{3}x\).
Write the final expression as a single logarithm: \(\ln\left(\text{simplified product}\right)\).
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Properties of Logarithms
Logarithmic properties allow the simplification and combination of logarithmic expressions. Key properties include 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 rules help rewrite multiple logarithmic terms as a single logarithm.
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Change of Base Property
Domain of Logarithmic Functions
The domain of a logarithmic function consists of all positive real numbers inside the logarithm. When combining logarithms, it is essential to ensure the resulting argument remains positive, as ln(x) is undefined for x ≤ 0. This affects the validity of the simplified expression.
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5:26Graphs of Logarithmic Functions
Algebraic Simplification
Algebraic simplification involves factoring and reducing expressions inside the logarithms before applying logarithmic properties. For example, factoring 3x² - 9x as 3x(x - 3) can make it easier to combine terms and simplify the overall expression.
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05:25Determine Continuity Algebraically
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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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Domanda del libro di testo
131. Let f(x) = x * e^(−x).
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