logx-ab-if-logx-a-30-andlogx-b-70-

Question Number 160939 by mathlove last updated on 09/Dec/21

l o g \underset{a b}{x} = ? i f l o g \underset{a}{x} = 30 a n d l o g \underset{b}{x} = 70

Commented by blackmamba last updated on 09/Dec/21

{\begin{cases} a = x^{\frac{1}{30}} \\ b = x^{\frac{1}{70}} \end{cases} \Rightarrow \log_{a b} (x) = \log_{x^{\frac{1}{30} + \frac{1}{70}}} (x) = \frac{1}{\frac{1}{30} + \frac{1}{70}} = 21

Answered by qaz last updated on 09/Dec/21

\log_{ab} x = \frac{lnx}{\ln (ab)} = \frac{lnx}{lna + lnb} = \frac{1}{\frac{lna}{lnx} + \frac{lnb}{lnx}} = \frac{1}{\frac{1}{\log_{a} x} + \frac{1}{\log_{b} x}} = \frac{1}{\frac{1}{30} + \frac{1}{70}} = 21

Answered by cortano last updated on 09/Dec/21

{\begin{cases} \log_{a} (x) = 30 \\ \log_{b} (x) = 70 \end{cases}

\log_{ab} (x) = \frac{1}{\log_{x} (ab)} = \frac{1}{\log_{x} (a) + \log_{x} (b)}

= \frac{1}{\frac{1}{30} + \frac{1}{70}} = \frac{30 \times 70}{100} = 21

Facebook Tweet Pin