Why is the graph of Y=2LNx "half" of the graph of

Y=LN(x squared)?

Perhaps I don't know what you mean by "half"

2ln(x) is the same as ln(x^2)

To understand why the graph of Y = 2ln(x) is considered "half" of the graph of Y = ln(x^2), let's examine the equations and their properties.

First, let's clarify that 2ln(x) is not the same as ln(x^2). However, there is a relationship between the two equations.

The equation Y = ln(x^2) represents the natural logarithm of the square of x. The square of x ensures that the values are always positive, as the square of any number is non-negative. This equation captures both positive and negative values of x, resulting in a symmetrical graph. In other words, the graph of Y = ln(x^2) is symmetric about the y-axis.

On the other hand, the equation Y = 2ln(x) represents twice the natural logarithm of x. This equation only captures positive values of x since the logarithm of negative numbers is undefined in the real number system. As a result, the graph of Y = 2ln(x) is only defined for x > 0 and covers only the right half of the Y = ln(x^2) graph.

To visually understand this, you can plot the two equations on a graphing calculator or software. You will observe that the graph of Y = 2ln(x) resembles the right half of the graph of Y = ln(x^2). This is why it is often described as "half" of the graph of Y = ln(x^2).