Textbook Question
Use the change-of-base theorem to find an approximation to four decimal places for each logarithm. See Example 8. . log_1/2 3
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6. Exponential & Logarithmic Functions
Properties of Logarithms
3:26 minutesProblem 87
Textbook Question
In Exercises 83–88, let logb 2 = A and logb 3 = C and Write each expression in terms of A and C.logb √(2/27)
Verified step by step guidance1
Start by expressing the square root as an exponent: \( \log_b \sqrt{\frac{2}{27}} = \log_b \left(\frac{2}{27}\right)^{1/2} \).
Use the power rule of logarithms, which states \( \log_b (x^n) = n \cdot \log_b x \), to rewrite the expression: \( \frac{1}{2} \cdot \log_b \left(\frac{2}{27}\right) \).
Apply the quotient rule of logarithms, \( \log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y \), to separate the terms: \( \frac{1}{2} \cdot (\log_b 2 - \log_b 27) \).
Recognize that \( \log_b 27 = \log_b (3^3) \) and use the power rule again: \( \log_b 27 = 3 \cdot \log_b 3 \).
Substitute the given values \( \log_b 2 = A \) and \( \log_b 3 = C \) into the expression: \( \frac{1}{2} \cdot (A - 3C) \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Logarithmic Properties
Logarithmic properties, such as the product, quotient, and power rules, are essential for manipulating logarithmic expressions. For instance, the quotient rule states that logb(m/n) = logb(m) - logb(n), allowing us to break down complex logarithmic expressions into simpler components.
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Change of Base Formula
The change of base formula allows us to express logarithms in terms of different bases. Specifically, logb(x) can be rewritten using natural logarithms or common logarithms, which is useful when we need to express logarithms in terms of known values, such as A and C in this problem.
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Square Roots and Exponents
Understanding square roots and their relationship with exponents is crucial in logarithmic expressions. The square root of a number can be expressed as that number raised to the power of 1/2, which can simplify the logarithmic expression further when combined with logarithmic properties.
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