solve:
$\displaystyle \log_{7} \left ( {\frac{1}
{\sqrt{7}}} \right )$
how would you go about this w/o a calculator
$\displaystyle 7^x = \frac{1}{\sqrt{7}}$
or
$\displaystyle \log_{7}1 - \log_{7}\sqrt{7} = x$
$\displaystyle \log_{7} \left ( {\frac{1}{\sqrt{7}}} \right ) = \log_{7}1 - \log_{7}\sqrt{7}$
$\displaystyle \log_{7}1 = 0$
$\displaystyle \log_{7} \left ( {\frac{1}{\sqrt{7}}} \right ) = - \log_{7}\sqrt{7}$
$\displaystyle \log_{7}\sqrt{7} = \log_{7}7^\frac{1}{2} = \frac{1}{2}\log_{7}7$
$\displaystyle \log_{7}7 = 1$
$\displaystyle \log_{7} \left ( {\frac{1}{\sqrt{7}}} \right ) = - \frac{1}{2}$