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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Hass 15th EditionCh. 7 - Transcendental Functions문제 7.2.3b
7장, 문제 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)

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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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.
추천 영상:
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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.
추천 영상:
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Graphs 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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