Why are log graphs of different equations not all regular parabolas?

[JBD2]

Can someone explain to me why the graph of:

logy=logx² is the graph of a regular parabola

logy=2logx is the graph of half a parabola (x>0)

log_xy=2 is the graph of half a parabola except x>0 and x cannot be equal to 1

I just don't understand why they're not all normal parabolas, and how the second two are different than the first.

 

[Tedjn]

(1) $\log y = \log x^2$ is a parabola.

(2) $\log y = 2\log x$ is equivalent to (1) only when x > 0, because otherwise the logarithm does not exist. That is, whenever x > 0, (1) = (2), but when x < 0, we actually have $\log y = 2\log(-x)$.

(3) $\log_x y = 2$. You get this from (2) by dividing by $\log x$. Thus, we already see that x > 0 from (2). Moreover, if x = 1, then $\log 1 = 0$, and you are dividing by 0, which is not allowed. Indeed, from (3), you see that if x = 1, then (3) can never equal 2 (1 to any power is still 1).