Let \(x \) and \(b\) be positive real numbers so that \(log_b(x^2) = 10\) Find \(log_{\sqrt[3]{b}} \left( \frac{1}{x} \right)\)
Let x and b be positive real numbers so that \(log_b(x^2) = 10. \) Find \(log_{\sqrt[3]{b}} \left( \frac{1}{x} \right). \)
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\(log_b(x^2) = 10 \)
\(b^{10}=x^2\\ \color{blue}x=b^5\)
\(log_{\sqrt[3]{b}} \left( \frac{1}{x} \right)\)
\( log_{\sqrt[3]{b}} \left( \frac{1}{b^5} \right)\\\)= 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
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