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Let \(x \) and \(b\) be positive real numbers so that \(log_b(x^2) = 10\) Find \(log_{\sqrt[3]{b}} \left( \frac{1}{x} \right)\)

 Apr 1, 2020
 #1
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Let x and b  be positive real numbers so that \(log_b(x^2) = 10. \)  Find \(log_{\sqrt[3]{b}} \left( \frac{1}{x} \right). \)

 

Hello Guest!

 

\(log_b(x^2) = 10 \)

 

\(b^{10}=x^2\\ \color{blue}x=b^5\)

 

\(log_{\sqrt[3]{b}} \left( \frac{1}{x} \right)\)

 

\( log_{\sqrt[3]{b}} \left( \frac{1}{b^5} \right)\\\)= z

\((\sqrt[3]{b})^z=b^{-5}\\ b^{\frac{z}{3}}=b^{-5}\\ \frac{z}{3}=-5\\ \color{blue}z=-15 \)

\( log_{\sqrt[3]{b}} \left( \frac{1}{b^5} \right)\\\)= z

\( log_{\sqrt[3]{b}} \left( \frac{1}{b^5} \right)\\\)= -15

laugh  !

 Apr 2, 2020
 #2
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Thank you so much!

Guest Apr 2, 2020
 #3
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Thank you for your thank you. It is rare here.

smiley  !

asinus  Apr 2, 2020

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