Textbook Question
Use a graphing utility and the change-of-base property to graph each function. y = log3 x
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6. Exponential & Logarithmic Functions
Properties of Logarithms
2:09 minutesProblem 83
Textbook Question
Use the change-of-base theorem to find an approximation to four decimal places for each logarithm. log1/2 3
Verified step by step guidance1
Identify the logarithm you need to approximate: \(\log_{\frac{1}{2}} 3\).
Recall the change-of-base formula: \(\log_a b = \frac{\log_c b}{\log_c a}\), 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 using base 10: \(\log_{\frac{1}{2}} 3 = \frac{\log_{10} 3}{\log_{10} \frac{1}{2}}\).
Use a calculator to find the values of \(\log_{10} 3\) and \(\log_{10} \frac{1}{2}\) separately, keeping several decimal places for accuracy.
Divide the value of \(\log_{10} 3\) by the value of \(\log_{10} \frac{1}{2}\) to get the approximate value of \(\log_{\frac{1}{2}} 3\), then round your answer to four decimal places.
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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 a logarithm with any base as a ratio 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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Properties of Logarithms
Understanding the properties of logarithms, such as how the base affects the value and the behavior of logarithmic functions, is essential. For example, logarithms with bases between 0 and 1 (like 1/2) decrease as the argument increases, which influences the sign and magnitude of the result.
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Decimal Approximation and Rounding
After computing the logarithm using the change-of-base formula, you must approximate the result to a specified number of decimal places. This involves using a calculator for precise values and rounding the final answer correctly to four decimal places to meet the problem's requirements.
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