Question

Daca
$log_{a}(27) = b$
atunci aratati ca
$log_{ \sqrt{3} }( \sqrt[6]{a} ) = \frac{1}{b}$
​

Answer 1

$\log_{a}27 = b \\ \\\\ \text{Formule:}\\\\ \log_{a}b^x =\log_{a^{\frac{1}{x}}}b \\ \log_{a^x}b = \log_{a}b^\frac{1}{x} \\ \log_{a}b = \dfrac{1}{\log_{b}a}\\ \\ \\\log_{\sqrt{3}}\sqrt[6]a =\log_{\sqrt 3}a^{\frac{1}{6}}=\log_{(\sqrt{3})^6}a = \log_{3^3}a =\\ \\ = \log_{27}a = \dfrac{1}{\log_{a}27} = \dfrac{1}{b}\quad (q.e.d.)$

Answer 2

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