How would I read an equation like this 6^(log_6 20)=20 ? How would I evaluate it? Why is it equal to 20?
In general $\displaystyle y=\ln(x) \Leftrightarrow e^{y}=x \Leftrightarrow e^{\ln(x)}=x$. Do you recognize this in your example?
You can also take the $\displaystyle \log_6$ of both sides and simplify with the logarithm rules.
I think I am beginning to. Why is it, though, that when the base of the logarithm in the exponent and the base of that logarithmic exponenent are equal, the answer is arguement? Is it because it is not an algebraic function, and is read differently because of that.
It's just a very handy proposition to work with logarithms. You'll see if you take the $\displaystyle \log_6$ of both sides then you get:
$\displaystyle \log_6(6^{\log_6(20)})=\log_6(20)$
$\displaystyle \Leftrightarrow \log_6(20)\cdot \log_6(6)=\log_6(20)$
$\displaystyle \Leftrightarrow \log_6(6)=1$
$\displaystyle \Leftrightarrow 6^1=6$
So indeed, they LHS and the RHS of the expression are equal.