Answer
(a) $x=\frac{1}{\log 6+1}$
(b) $x\approx 0.562382$
Work Step by Step
(a) We need to solve:
$10^{1-x}=6^{x}$
We take the log of both sides and use log rules to simplify:
$\log 10^{1-x}=\log 6^{x}$
$1-x=x(\log 6)$
$1=x(\log 6)+x$
$1=x(\log 6+1)$
$x=\frac{1}{\log 6+1}$
(b) We solve using a calculator:
$x\approx 0.562382$