Textbook Question
Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. logπ 63
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6. Exponential & Logarithmic Functions
Properties of Logarithms
2:24 minutesProblem 81
Textbook Question
Use the change-of-base theorem to find an approximation to four decimal places for each logarithm. log8 0.59
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 (commonly 10 or \(e\)).
Apply the change-of-base formula to \(\log_8 0.59\) by choosing base 10 (common logarithm): \(\log_8 0.59 = \frac{\log_{10} 0.59}{\log_{10} 8}\).
Use a calculator to find the values of \(\log_{10} 0.59\) and \(\log_{10} 8\) separately (do not calculate the final division yet).
Divide the value of \(\log_{10} 0.59\) by the value of \(\log_{10} 8\) to get the approximate value of \(\log_8 0.59\).
Round the result to four decimal places to complete the approximation.
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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 have log or ln functions.
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Change of Base Property
Logarithm Properties
Logarithms are the inverse operations of exponentiation and follow specific properties such as log_b(xy) = log_b(x) + log_b(y) and log_b(x^r) = r log_b(x). Understanding these properties helps simplify expressions and solve logarithmic equations effectively.
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Decimal Approximation and Rounding
When calculating logarithms, results often come as decimal numbers that need to be approximated to a certain number of decimal places. Rounding to four decimal places means adjusting the number so that only four digits appear after the decimal point, ensuring precision and clarity in answers.
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