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Determine whether the statement is true or false for all x > 0, y > 0.

Determine whether the statement is true or false for all x > 0, y > 0.

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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**Determine whether the statement is true or false for all \( x > 0, y > 0 \):** \[ \frac{\log_b x}{\log_b y} = \log_b x - \log_b y \] - ○ True - ○ False **If it is false, write an example that disproves the statement.** - ○ The statement is true. - ○ \( \log 100 - \log 10 = -1\) but \( \frac{\log 100}{\log 10} = \frac{1}{2} \) - ○ \( \frac{\log 100}{\log 10} = \frac{10}{1} = 10\) but \(\log 100 - \log 10 = 9 \) - ○ \( \frac{\log 100}{\log 10} = \frac{2}{1} = 2 \) but \(\log 100 - \log 10 = 1 \) - ○ \( \log 100 - \log 10 = 90 \) but \( \frac{\log 100}{\log 10} = \frac{100}{10} = 10 \)
Transcribed Image Text:**Determine whether the statement is true or false for all \( x > 0, y > 0 \):** \[ \frac{\log_b x}{\log_b y} = \log_b x - \log_b y \] - ○ True - ○ False **If it is false, write an example that disproves the statement.** - ○ The statement is true. - ○ \( \log 100 - \log 10 = -1\) but \( \frac{\log 100}{\log 10} = \frac{1}{2} \) - ○ \( \frac{\log 100}{\log 10} = \frac{10}{1} = 10\) but \(\log 100 - \log 10 = 9 \) - ○ \( \frac{\log 100}{\log 10} = \frac{2}{1} = 2 \) but \(\log 100 - \log 10 = 1 \) - ○ \( \log 100 - \log 10 = 90 \) but \( \frac{\log 100}{\log 10} = \frac{100}{10} = 10 \)
**Title: Understanding Logarithmic Properties** Determine if the statement is true or false for all \( x > 0, y > 0 \). \[ \log_b(x + y) = \log_b x + \log_b y \] - ○ True - ○ False **If it is false, write an example that disproves the statement.** - ○ The statement is true. - ○ \( \log(10 + 10) = \log 20 = 2 \) but \( \log 10 + \log 10 \neq 2 \) - ○ \( \log(10 + 10) = \log 20 = 10 \) but \( \log 10 + \log 10 \neq 10 \) - ○ \( \log 10 + \log 10 = 2 \) but \( \log(10 + 10) = \log 20 \neq 2 \) - ○ \( \log 10 + \log 10 = 10 \) but \( \log(10 + 10) = \log 20 \neq 10 \) **Explanation:** This exercise involves evaluating the properties of logarithms. The statement suggests a property similar to the multiplication property of logarithms, which is actually incorrect. Logarithms do not distribute over addition. The list provides example options that illustrate why the equation \(\log_b(x + y) = \log_b x + \log_b y\) might be false by showing different calculations and their results.
Transcribed Image Text:**Title: Understanding Logarithmic Properties** Determine if the statement is true or false for all \( x > 0, y > 0 \). \[ \log_b(x + y) = \log_b x + \log_b y \] - ○ True - ○ False **If it is false, write an example that disproves the statement.** - ○ The statement is true. - ○ \( \log(10 + 10) = \log 20 = 2 \) but \( \log 10 + \log 10 \neq 2 \) - ○ \( \log(10 + 10) = \log 20 = 10 \) but \( \log 10 + \log 10 \neq 10 \) - ○ \( \log 10 + \log 10 = 2 \) but \( \log(10 + 10) = \log 20 \neq 2 \) - ○ \( \log 10 + \log 10 = 10 \) but \( \log(10 + 10) = \log 20 \neq 10 \) **Explanation:** This exercise involves evaluating the properties of logarithms. The statement suggests a property similar to the multiplication property of logarithms, which is actually incorrect. Logarithms do not distribute over addition. The list provides example options that illustrate why the equation \(\log_b(x + y) = \log_b x + \log_b y\) might be false by showing different calculations and their results.
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