Textbook Question
Solve each equation. Round answers to the nearest hundredth as needed. x2/3 =36
Ch. 4 - Inverse, Exponential, and Logarithmic Functions
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Table of contents
Chapter 5, Problem 6
Answer each of the following. Write log3 12 in terms of natural logarithms using the change-of-base theorem.
Verified step by step guidance1
Recall the change-of-base theorem, which states that for any positive numbers \(a\), \(b\), and base \(c\) (where \(a \neq 1\) and \(c \neq 1\)), the logarithm \(\log_a b\) can be rewritten as \(\frac{\log_c b}{\log_c a}\).
In this problem, we want to express \(\log_3 12\) in terms of natural logarithms, which means using the natural logarithm function \(\ln\) as the new base.
Apply the change-of-base formula with \(a = 3\), \(b = 12\), and \(c = e\) (the base of natural logarithms), so \(\log_3 12 = \frac{\ln 12}{\ln 3}\).
Write the expression clearly as \(\log_3 12 = \frac{\ln 12}{\ln 3}\), which is the logarithm base 3 of 12 expressed in terms of natural logarithms.
This expression can now be used for further calculations or evaluations using a calculator that has the natural logarithm function.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Logarithms
A logarithm answers the question: to what exponent must the base be raised to produce a given number? For example, log_b(a) is the exponent x such that b^x = a. Understanding this definition is essential for manipulating and converting logarithmic expressions.
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Logarithms Introduction
Change-of-Base Theorem
The change-of-base theorem allows you to rewrite a logarithm with any base in terms of logarithms with a different base. It states that log_b(a) = log_c(a) / log_c(b), where c is a new base, often chosen as e (natural logarithm) or 10 for convenience.
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5:36Change of Base Property
Natural Logarithms
Natural logarithms use the base e (approximately 2.718) and are denoted as ln(x). They are widely used in calculus and higher mathematics. Expressing logarithms in terms of natural logs simplifies calculations and connects logarithmic functions to exponential growth and decay.
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Related Practice
Textbook Question
Fill in the blank(s) to correctly complete each sentence. If ƒ(x) = 4x, then ƒ(2) = and ƒ(-2) = ________.
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Textbook Question
Solve each equation. Round answers to the nearest hundredth as needed. (1/4)x=64
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Textbook Question
Answer each of the following. Between what two consecutive integers must log2 12 lie?
Textbook Question
Fill in the blank(s) to correctly complete each sentence. The graph of ƒ(x) = -(1/3)x+4-5 is that of ƒ(x) = (1/3)x reflected across the ______ -axis, translated to the left ______ units and down _______ units.