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Ch. 1 - Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Briggs 3rd EditionCh. 1 - Functions문제 1.49
1장, 문제 1.49

Properties of logarithms Assume logbx = 0.36, logby= 0.56 and logbz = 0.83 . Evaluate the following expressions.


logb (√x) / (³√z)

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Start by using the properties of logarithms to simplify the expression \( \log_b \left( \frac{\sqrt{x}}{\sqrt[3]{z}} \right) \).
Apply the quotient rule of logarithms: \( \log_b \left( \frac{\sqrt{x}}{\sqrt[3]{z}} \right) = \log_b (\sqrt{x}) - \log_b (\sqrt[3]{z}) \).
Use the power rule of logarithms: \( \log_b (\sqrt{x}) = \log_b (x^{1/2}) = \frac{1}{2} \log_b x \).
Similarly, apply the power rule to \( \log_b (\sqrt[3]{z}) = \log_b (z^{1/3}) = \frac{1}{3} \log_b z \).
Substitute the given values: \( \frac{1}{2} \log_b x = \frac{1}{2} \times 0.36 \) and \( \frac{1}{3} \log_b z = \frac{1}{3} \times 0.83 \), then subtract the two results.

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주요 개념

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Properties of Logarithms

The properties of logarithms include rules that simplify the manipulation of logarithmic expressions. Key properties include the product rule (logb(mn) = logb(m) + logb(n)), the quotient rule (logb(m/n) = logb(m) - logb(n)), and the power rule (logb(m^k) = k * logb(m)). Understanding these properties is essential for evaluating complex logarithmic expressions.
추천 영상:
05:36
Change of Base Property

Change of Base Formula

The change of base formula allows the conversion of logarithms from one base to another, expressed as logb(a) = logk(a) / logk(b) for any positive k. This is particularly useful when dealing with logarithms of different bases, enabling easier calculations and comparisons. It is important for evaluating logarithmic expressions when the base is not easily manageable.
추천 영상:
05:36
Change of Base Property

Radicals and Exponents

Radicals, such as square roots and cube roots, can be expressed in terms of exponents, where √x = x^(1/2) and ³√z = z^(1/3). This relationship is crucial when simplifying logarithmic expressions involving roots, as it allows the application of the power rule of logarithms. Recognizing this connection helps in transforming and evaluating logarithmic expressions effectively.
추천 영상:
7:39
Introduction to Exponent Rules
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