I just now know what LN is, natural log. But I still don't quite understand how to use it.
Let's say that I have a problem :

6^x=1500

I know that to get what x equals you have to use LN. I still get confuse with LN, LOG, and LG^2 and the rest. How do I know which to use when?

This is how I though I should solve to above problem:
LOG^6 x=1500

x=LN(1500)/ln(4)

IS THAT RIGHT?

For the type of equation that you're specifying, it doesn't really matter whether you use LN, LOG, or a logarithm to another base.

6^x=1500

So

LOG(6^x) = LOG(1500)
and
LN (6^x) = LN(1500)

then
x LOG(6) = LOG(1500)
and
x LN(6) = LN(1500)

finally
x = LOG(1500) / LOG(6)
and
x = LN(1500) / LN(6)

I'm not sure where you got the ln(4).

If you had an equation with LN or LOG in it, and you have to solve for something inside the LN() or LOG(), you have to take the appropriate antilog (e^x) or (10^x).

There are equations in physics and chemistry that require LN or LOG. That's why these appear.