Logarithms

What is the value of $$[\log_{10}(5 \cdot \log_{10} 100)]^2$$? ...

If $$ \log_x (0.1) = -\frac{1}{3} $$, find the value of $$ x $$. ...

Determine the value of $$ \log_2 \log_2 \log_3 \log_3 27^3 $$. ...

If $$ \log 3 $$, $$ \log (3^x - 2) $$, and $$ \log (3^x + 4) $$ are in arithmetic progression, find the value of $$ x $...

If $$\log x + \log y = \log (x + y)$$, find the relationship between $$x$$ and $$y$$. ...