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Ch. 4 - Inverse, Exponential, and Logarithmic Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 4 - Inverse, Exponential, and Logarithmic FunctionsProblem 87
Chapter 5, Problem 87

Use the change-of-base theorem to find an approximation to four decimal places for each logarithm. log√13 12

Verified step by step guidance
1
Identify the logarithm you need to approximate: \(\log_{\sqrt{13}} 12\).
Recall the change-of-base formula: \(\log_a b = \frac{\log_c b}{\log_c a}\), where \(c\) is any positive number (commonly 10 or \(e\)).
Apply the change-of-base formula using base 10 (common logarithm): \(\log_{\sqrt{13}} 12 = \frac{\log_{10} 12}{\log_{10} \sqrt{13}}\).
Express \(\sqrt{13}\) as an exponent: \(\sqrt{13} = 13^{1/2}\), so \(\log_{10} \sqrt{13} = \log_{10} 13^{1/2} = \frac{1}{2} \log_{10} 13\).
Calculate the values of \(\log_{10} 12\) and \(\log_{10} 13\), then substitute back into the fraction and simplify to find the approximate value.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Change-of-Base Theorem

The change-of-base theorem allows you to rewrite logarithms with any base into a quotient of logarithms with a new base, typically base 10 or e. It states that log_b(a) = log_c(a) / log_c(b), where c is the new base. This is useful for calculating logarithms on calculators that only support common or natural logs.
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Logarithms with Radical Bases

Logarithms can have bases that are radicals, such as √13. Understanding how to handle these involves recognizing that the base can be expressed as an exponent (e.g., √13 = 13^(1/2)), which helps in applying logarithm properties and simplifying expressions.
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Expanding Radicals

Rounding and Approximation

When calculating logarithms numerically, the result is often an irrational number. Approximating to a certain number of decimal places, such as four, requires rounding the decimal expansion carefully to ensure accuracy and precision in the final answer.
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Related Practice
Textbook Question

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