Question

Va rog ajutati ma sa demonstrez egalitatea

Answer 1

$\textbf{Formule:}\\ \\ \log_{a^x}b = \log_{a}b^{\frac{1}{x}}\\ \\ \log_{a}b^x = \log_{a^{\frac{1}{x}}}b\\\\\log_{a}b = \dfrac{1}{\log_{b}a}$

$\textbf{Demonstratie:}\\\\\log_{\frac{b}{a}}(\log_{a}b) = \log_{(\frac{a}{b})^{-1}}(\log_{a}b) =\\ \\ =\log_{\frac{a}{b}}(\log_{a}b)^{\frac{1}{-1}}= \log_{\frac{a}{b}}(\log_{a}b)^{-1} =\\ \\ = \log_{\frac{a}{b}}\left(\frac{1}{\log_{a}b}\right) = \log_{\frac{a}{b}}(\log_{b}a)\quad \checkmark$