Answer
(a) $101,1.1$
(b) $64$
(c) $1.585$
Work Step by Step
(a) Take log of each side $\log((x-1)^{\log(x-1)})=\log(100(x-1))$ we have
$(\log(x-1))^2=\log100+\log(x-1)$, let $y=\log(x-1)$,
$(\log(x-1))^2-\log(x-1)-\log100=0$, let $y=\log(x-1)$,
$y^2-y-2=0, y=2,-1$
$\log(x-1)=2, x-1=10^2, x=101$
$\log(x-1)=-1, x-1=1/10, x=1.1$
(b) $\frac{\log x}{\log 2}+\frac{\log x}{\log4}+\frac{\log x}{\log 8}=11$
$(1+1/2+1/3)\frac{\log x}{\log2}=11$ (using $\log8=\log2^3=3\log2$)
$\log x=11\times\frac{6}{11}\log2=\log2^6$
$x=2^6=64$
(c) $2^{2x}-2\cdot2^x-3=0$, let $y=2^x$, we have
$y^2-2y-3=0, y=3, -1$ (discard -1 as $y>0$)
$2^x=3, x=\log_23=1.585$
