Textbook Question
Use the change-of-base theorem to find an approximation to four decimal places for each logarithm. log1/2 3
Ch. 4 - Inverse, Exponential, and Logarithmic Functions
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Table of contents
Chapter 5, Problem 85
Use the change-of-base theorem to find an approximation to four decimal places for each logarithm. logπ e
Verified step by step guidance1
Recall the change-of-base formula for logarithms: \(\log_a b = \frac{\log_c b}{\log_c a}\), where \(c\) is any positive number different from 1. Common choices for \(c\) are 10 (common logarithm) or \(e\) (natural logarithm).
Apply the change-of-base formula to \(\log_{\pi} e\) by choosing the natural logarithm base \(e\): \(\log_{\pi} e = \frac{\ln e}{\ln \pi}\).
Evaluate the numerator \(\ln e\). Since the natural logarithm of \(e\) is 1, this simplifies the expression to \(\frac{1}{\ln \pi}\).
Calculate \(\ln \pi\) using a calculator or logarithm table to get a decimal approximation.
Divide 1 by the decimal approximation of \(\ln \pi\) to find the approximate value of \(\log_{\pi} e\), then round your answer to four decimal places.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Logarithms and Their Bases
A logarithm answers the question: to what power must the base be raised to produce a given number? The notation log_b a means the logarithm of a with base b. Understanding how different bases affect the value of logarithms is essential for manipulating and approximating them.
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Logarithms Introduction
Change-of-Base Theorem
The change-of-base theorem allows you to rewrite a logarithm with any base into a ratio of logarithms with a more convenient base, typically base 10 or e. It states that log_b a = (log_c a) / (log_c b), where c is a new base. This is useful for calculating logarithms on calculators that only support certain bases.
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5:36Change of Base Property
Approximation of Logarithms Using Calculators
Since calculators usually provide logarithms in base 10 or base e (natural logs), approximating logarithms with other bases requires using the change-of-base formula. After rewriting, you compute the numerator and denominator logarithms and divide, then round the result to the desired decimal places.
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Related Practice
Textbook Question
Solve each equation. See Examples 4–6. x2/3 = 4
Textbook Question
Use the change-of-base theorem to find an approximation to four decimal places for each logarithm. log√13 12
Textbook Question
Solve each equation. x5/2 = 32
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Textbook Question
Solve each equation. Give solutions in exact form. log x2 = (log x)2
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Textbook Question
Solve each equation for the indicated variable. Use logarithms with the appropriate bases. p = a + (k/ln x), for x
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