1. Log10²x+log10x²=log10² 2-1

Question

2. Log4(log2x)+log2(log4x)=2
3. X^logx+5/3= 10^5+log x
4. Log 1/2(x-1)+ log 1/2(x+1)-log1/√2(7-x)=1

Answer 1

Again, having a hard time trying to establish the order of operation.
e.g. #3
I put it into Wolfram the way you typed it , and the webpage interpreted it the same way I did

http://www.wolframalpha.com/input/?i=+X%5Elogx%2B5%2F3%3D+10%5E5%2Blog+x

What a messy equation that would be to solve
I have a feeling you meant something like

X^logx + 5/3= 10^(5+log x)

looks like very messy stuff for the other equations as well, the way you typed it.