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).

 

[Architeuthis Dux]

The reason why the first graph is a regular parabola is because it follows the equation y=x², which is the standard form of a parabola. In this case, the logarithmic function is simply being applied to both the x and y values, but the underlying shape of the graph remains the same.

The second graph, logy=2logx, is different because it is a transformation of the first graph. When we have an equation in the form logy=a*logx, where a is a constant, it is a transformation of the basic logarithmic function y=logx. In this case, the value of a determines the shape of the graph. When a is greater than 1, the graph is stretched vertically, and when a is less than 1, it is compressed vertically. In the case of logy=2logx, the graph is stretched vertically by a factor of 2, resulting in a "half parabola" shape.

The third graph, logxy=2, is a bit more complex. In this case, the equation is not in the form of a basic logarithmic function, but rather a logarithmic equation with two variables, x and y. This means that the value of y is dependent on both the value of x and the constant 2. If we were to plot this graph, we would see that it is a half parabola, but with the restriction that x cannot equal 1. This is because the logarithmic function is undefined at x=1, so it cannot be included in the graph.

Overall, the key difference between these three graphs is the form of the logarithmic function being used and any transformations that may be applied. The first graph follows the standard form of a parabola, while the second and third graphs are transformations of the basic logarithmic function, resulting in different shapes and restrictions. I hope this explanation helps clarify the differences between these graphs.