Textbook Question
Working with composite functions
Find possible choices for outer and inner functions ƒ and g such that the given function h equals ƒ o g.
h(x) = (2) / ( x⁶ + x² + 1)²
Ch. 1 - Functions
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- 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: 9780136847243Not the one you use?Change textbook
Chapter 1, Problem 1.3.46
Properties of logarithms Assume logbx = 0.36, logby= 0.56 and logbz = 0.83 . Evaluate the following expressions.
logbx²
Verified step by step guidance1
Recognize that the expression \( \log_b x^2 \) can be simplified using the power rule of logarithms, which states \( \log_b (x^n) = n \cdot \log_b x \).
Apply the power rule to the expression: \( \log_b x^2 = 2 \cdot \log_b x \).
Substitute the given value of \( \log_b x = 0.36 \) into the expression: \( 2 \cdot 0.36 \).
Perform the multiplication to simplify the expression: \( 2 \cdot 0.36 \).
The result of the multiplication gives the value of \( \log_b x^2 \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Properties of Logarithms
Logarithms have several key properties that simplify calculations. The most relevant ones include the product property (log_b(mn) = log_b(m) + log_b(n)), the quotient property (log_b(m/n) = log_b(m) - log_b(n)), and the power property (log_b(m^n) = n * log_b(m)). Understanding these properties is essential for manipulating logarithmic expressions effectively.
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Change of Base Property
Power Property of Logarithms
The power property states that the logarithm of a number raised to an exponent can be expressed as the exponent multiplied by the logarithm of the base number. For example, log_b(x^n) = n * log_b(x). This property is particularly useful when evaluating expressions involving powers, as it allows for simplification before calculation.
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Base of Logarithm
The base of a logarithm indicates the number that is raised to a power to obtain a given value. In the expression log_b(x), 'b' is the base. Understanding the base is crucial because it affects the value of the logarithm and the application of logarithmic properties. In this question, knowing the base helps in evaluating log_b(x^2) using the power property.
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Related Practice
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a. log₁₀ 1000
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