Domanda del libro di testo
Suppose ƒ is an even function with ƒ(2) = 2 and g is an odd function with g(2) = -2. Evaluate ƒ(-2) , ƒ(g(2)), and g(ƒ(-2))
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Ch. 1 - Functions
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 1, Problema 1.49
Properties of logarithms Assume logbx = 0.36, logby= 0.56 and logbz = 0.83 . Evaluate the following expressions.
logb (√x) / (³√z)
Guida verificata passo dopo passo1
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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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.
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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.
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05:36Change 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.
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7:39Introduction to Exponent Rules
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