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04 loga(u) oga(EK 43).

04 loga(u) oga(EK 43).

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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**Logarithmic Equation Example** Given the expression: \[ 4 \log_{a} (6x) = \frac{1}{2} \log_{a} (5x + 3) \] Explanation: 1. **Original Equation**: - The given logarithmic equation involves two logarithmic expressions set equal to each other. - On the left side of the equation, the logarithm of \( 6x \) is multiplied by 4. - On the right side of the equation, the logarithm of \( 5x + 3 \) is multiplied by \(\frac{1}{2}\). 2. **Logarithmic Properties**: - Recall the property that \( k \log_{b}(M) = \log_{b}(M^k) \). - Using this property, we can transform both sides of the equation. 3. **Transformed Equation**: - For the left side: \( 4 \log_{a} (6x) = \log_{a} ((6x)^4) = \log_{a} (6^4 x^4) \). - For the right side: \( \frac{1}{2} \log_{a} (5x + 3) = \log_{a} ((5x + 3)^{1/2}) = \log_{a} (\sqrt{5x + 3}) \). By understanding and applying logarithmic properties, complex equations like the one above can be simplified and solved methodically. Make sure to verify the permissible domain values for the variables involved when dealing with logarithms.
Transcribed Image Text:**Logarithmic Equation Example** Given the expression: \[ 4 \log_{a} (6x) = \frac{1}{2} \log_{a} (5x + 3) \] Explanation: 1. **Original Equation**: - The given logarithmic equation involves two logarithmic expressions set equal to each other. - On the left side of the equation, the logarithm of \( 6x \) is multiplied by 4. - On the right side of the equation, the logarithm of \( 5x + 3 \) is multiplied by \(\frac{1}{2}\). 2. **Logarithmic Properties**: - Recall the property that \( k \log_{b}(M) = \log_{b}(M^k) \). - Using this property, we can transform both sides of the equation. 3. **Transformed Equation**: - For the left side: \( 4 \log_{a} (6x) = \log_{a} ((6x)^4) = \log_{a} (6^4 x^4) \). - For the right side: \( \frac{1}{2} \log_{a} (5x + 3) = \log_{a} ((5x + 3)^{1/2}) = \log_{a} (\sqrt{5x + 3}) \). By understanding and applying logarithmic properties, complex equations like the one above can be simplified and solved methodically. Make sure to verify the permissible domain values for the variables involved when dealing with logarithms.
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