Textbook Question
Use the various properties of exponential and logarithmic functions to evaluate the expressions in parts (a)–(c). Given ƒ(x) = log2 x, find ƒ(22 log_2 2)
Ch. 4 - Inverse, Exponential, and Logarithmic Functions
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Table of contents
Chapter 5, Problem 99
Work each problem. Which of the following is equivalent to 2 ln(3x) for x > 0?
A. ln 9 + ln x
B. ln 6x
C. ln 6 + ln x
D. ln 9x2
Verified step by step guidance1
Recall the logarithmic property that states: \(a \ln b = \ln b^a\). This means you can rewrite \(2 \ln(3x)\) as \(\ln((3x)^2)\).
Apply the exponent inside the logarithm: \((3x)^2 = 3^2 \times x^2 = 9x^2\). So, \(2 \ln(3x) = \ln(9x^2)\).
Recognize that \(\ln(9x^2)\) can be separated using the logarithm product rule: \(\ln(ab) = \ln a + \ln b\). Therefore, \(\ln(9x^2) = \ln 9 + \ln x^2\).
Use the power rule for logarithms on \(\ln x^2\): \(\ln x^2 = 2 \ln x\). So, \(\ln(9x^2) = \ln 9 + 2 \ln x\).
Compare the expression \(\ln(9x^2)\) with the given options to identify which one matches \(2 \ln(3x)\).
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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 specific properties that simplify expressions, such as the product rule (ln a + ln b = ln(ab)) and the power rule (k ln a = ln(a^k)). These rules allow rewriting complex logarithmic expressions into simpler or equivalent forms.
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Change of Base Property
Power Rule for Logarithms
The power rule states that multiplying a logarithm by a constant is equivalent to taking the logarithm of the argument raised to that constant: k ln(a) = ln(a^k). This is essential for rewriting expressions like 2 ln(3x) as ln((3x)^2).
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Guided course
5:50Power Rules
Domain Restrictions in Logarithmic Functions
Logarithmic functions are defined only for positive arguments. For ln(3x), the domain restriction x > 0 ensures the argument 3x is positive, which is necessary for the expression to be valid and for applying logarithmic properties correctly.
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3:51Domain Restrictions of Composed Functions
Related Practice
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Textbook Question
Use the various properties of exponential and logarithmic functions to evaluate the expressions in parts (a)–(c). Given ƒ(x) = log2 x, find ƒ(2log_2 2)
Textbook Question
Use the various properties of exponential and logarithmic functions to evaluate the expressions in parts (a)–(c). Given ƒ(x) = log2 x, find ƒ(27)