Textbook Question
Use the change-of-base theorem to find an approximation to four decimal places for each logarithm. logπ e
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6. Exponential & Logarithmic Functions
Properties of Logarithms
2:42 minutesProblem 87
Textbook Question
Use the change-of-base theorem to find an approximation to four decimal places for each logarithm. log√13 12
Verified step by step guidance1
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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Video duration:
2mKey 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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Change of Base Property
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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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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