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Let a,b, and c be the roots of 24x^3 - 121x^2 + 87x - 8 = 0 Find $\log_3(a)+\log_3(b)+\log_3(c).

 May 12, 2021
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Let a,b, and c be the roots of \(24x^3 - 121x^2 + 87x - 8\) = 0
Find
\( \log_3(a)+\log_3(b)+\log_3(c)\).

 

Vieta: \(-abc = -\dfrac{8}{24}\)

 

\(\begin{array}{|rcll|} \hline && \mathbf{\log_3(a)+\log_3(b)+\log_3(c)} \\\\ &=& \log_3(abc) \\\\ &=& \dfrac{\log(abc)}{\log(3)} \quad | \quad -\dfrac{8}{24}=-abc \\\\ &=& \dfrac{\log \left(\dfrac{8}{24} \right)}{\log(3)} \\\\ &=& \dfrac{\log \left(\dfrac{1}{3} \right)}{\log(3)} \\\\ &=& \dfrac{ \log(1)-\log(3) }{\log(3)} \quad | \quad \log(1) = 0\\\\ &=& \dfrac{ -\log(3) }{\log(3)} \\\\ &=& \mathbf{-1} \\ \hline \end{array}\)

 

laugh

 May 12, 2021

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