$\displaystyle \frac{1}{2}mv^2$
$\displaystyle 20\:\log.\left(\frac{2x10^-1}{2x10^-5}\right)\:=\:80$
10\:\log.\left[\left(\frac{2x10^-1}^2{2x10^-5}^2\right)\right]\:=\:80
where did i go wrong on equation 3, i have deleted the math tags to show work, i was trying to get the large outside brackets [ ]
10 log.[(2x10^-1)^2/(2x10^-5)^2]
also say if i want to show the multiply sign should i use x or *
Either 10\:\log.\left[\left(\frac{2\times 10^{-1}}{2\times 10^{-5}}\right)^2\right]\:=\:80, giving $\displaystyle 10\:\log.\left[\left(\frac{2\times 10^{-1}}{2\times 10^{-5}}\right)^2\right]\:=\:80$,
or 10\:\log.\left[\left(\frac{(2\times 10^{-1})^2}{(2\times 10^{-5})^2}\right)\right]\:=\:80, giving $\displaystyle 10\:\log.\left[\left(\frac{(2\times 10^{-1})^2}{(2\times 10^{-5})^2}\right)\right]\:=\:80$,
will work. But you can't use braces { } to do the same job as parentheses ( ). They have completely different functions in TeX.
For the multiplication sign, use \times.
Notice also that if an exponent consists of more than one symbol then you need to enclose the symbols in braces. Otherwise only the first symbol will be superscripted. For example, 10^-1 gives $\displaystyle 10^-1$, but 10^{-1} gives $\displaystyle 10^{-1}$.
I'll try again:
I hope i get better this time?!
10\:log.\:\left[\left(\frac{(2\times 10^{-1})^3}{2\times10^{-5})^3}\right)\right]\:=\:30\:log.\left(\frac{2\times10^{-1}){2\times10^{-5})\right)\:=120
Probably because what you wrote was "[tex]10\:log.\left[\left(\frac{2\times10^{-1}}{2\times10^{-5}}\right)^3\right]\:=\:30\:log.\left(\frac{2\times10^{-1}}{2\times10^{-5}}\right)\:=\:80[/tex]\:=\:80", with an additional "\:=\:80" at the end.
While I'm about it, here are a couple of extra comments. You don't normally put a full stop after "log", and in TeX you should write "\log". The backslash converts the letters from italic to roman, which looks better (the same applies to other common functions: \cos, \sin, \exp, ...).
Also, you shouldn't normally need to put extra space in math expressions by inserting spaces like "\:". TeX is designed to incorporate good spacing automatically.