[b](a) [/b] Find the minimal distance between the points of the graph of the function $y=\ln x$ from the line $y=x$.

[b](b)[/b] Find the minimal distance between two points, one of the point is in the graph of the function $y=e^x$ and the other  point in the graph of the function $y=ln x$.
**(a)** Find the minimal distance between the points of the graph of the function \( y = \ln x \) from the line \( y = x \).

1. To find the minimal distance between the curve \( y = \ln x \) and the line \( y = x \), we need to minimize the distance function between a point \((x, \ln x)\) on the curve and a point \((x, x)\) on the line. The distance \( D \) between these points is given by:
   \[
   D = \sqrt{(x - x)^2 + (\ln x - x)^2} = \sqrt{(\ln x - x)^2}
   \]
   Simplifying, we get:
   \[
   D = |\ln x - x|
   \]

2. To find the minimum distance, we need to minimize the function \( f(x) = |\ln x - x| \). Since \( \ln x \) is a continuous and differentiable function for \( x > 0 \), we can find the critical points by setting the derivative of \( \ln x - x \) to zero:
   \[
   \frac{d}{dx} (\ln x - x) = \frac{1}{x} - 1 = 0
   \]
   Solving for \( x \):
   \[
   \frac{1}{x} = 1 \implies x = 1
   \]

3. At \( x = 1 \), we evaluate the distance:
   \[
   D = |\ln 1 - 1| = |0 - 1| = 1
   \]

4. The minimal distance between the points of the graph of \( y = \ln x \) and the line \( y = x \) is:
   \[
   \boxed{1}
   \]

**(b)** Find the minimal distance between two points, one of the points is on the graph of the function \( y = e^x \) and the other point is on the graph of the function \( y = \ln x \).

1. To find the minimal distance between the graphs of \( y = e^x \) and \( y = \ln x \), we consider the symmetry of the graphs about the line \( y = x \). The minimal distance between these graphs will be twice the minimal distance found in part (a).

2. From part (a), the minimal distance between the graph of \( y = \ln x \) and the line \( y = x \) is 1. Therefore, the minimal distance between the graphs of \( y = e^x \) and \( y = \ln x \) is:
   \[
   2 \times 1 = 2
   \]

3. The minimal distance between the graphs of \( y = e^x \) and \( y = \ln x \) is:
   \[
   \boxed{2}
   \]