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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 7.2.1d
7장, 문제 7.2.1d

1. Express the following logarithms in terms of ln 2 and ln 3.
d. ln ∛9

검증된 단계별 안내
1
Recall that the cube root of 9 can be expressed as an exponent: \(\sqrt[3]{9} = 9^{\frac{1}{3}}\).
Rewrite the logarithm using the exponent rule: \(\ln \sqrt[3]{9} = \ln \left(9^{\frac{1}{3}}\right) = \frac{1}{3} \ln 9\).
Express 9 as a power of 3: \(9 = 3^2\).
Use the logarithm power rule to rewrite \(\ln 9\): \(\ln 9 = \ln \left(3^2\right) = 2 \ln 3\).
Substitute back to get the expression in terms of \(\ln 3\): \(\ln \sqrt[3]{9} = \frac{1}{3} \times 2 \ln 3 = \frac{2}{3} \ln 3\).

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주요 개념

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

Properties of Logarithms

Logarithmic properties such as the power rule (ln(a^b) = b ln(a)) and product rule allow us to rewrite complex logarithms in simpler forms. These rules help express logarithms of roots and products in terms of basic logarithms.
추천 영상:
05:36
Change of Base Property

Natural Logarithm (ln)

The natural logarithm, denoted ln, is the logarithm with base e. Understanding ln and its behavior is essential for manipulating expressions involving logarithms, especially when converting between different bases or simplifying expressions.
추천 영상:
05:18
Derivative of the Natural Logarithmic Function

Expressing Numbers as Powers or Products

To rewrite logarithms in terms of ln 2 and ln 3, numbers like 9 must be expressed as powers or products of 2 and 3. For example, 9 = 3^2, which allows the use of logarithm properties to break down ln(∛9) into terms involving ln 3.
추천 영상:
04:21
The Product Rule Example 1
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교과서 질문

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