The minimal distance between the curves defined by the equations \(y=e^x\) and \(y=\ln x\) is determined to be \(\sqrt{2}\). This distance occurs at the points where the tangents to the graphs are parallel to the line \(y=x\), specifically at the coordinates \((0,1)\) for the exponential function and \((1,0)\) for the logarithmic function. The relationship between these functions reveals that they are mirror images of each other across the line \(y=x\).
PREREQUISITESMathematics students, educators, and anyone interested in the geometric relationships between exponential and logarithmic functions.
The graphs of those two functions are mirror images of each other in the line $y=x$. So the minimal distance between them will be the distance between the points where they come closest to that line. That will occur at the points where the tangents to the graphs are parallel to the line (which has slope $1$, of course), in other words at the points $(0,1)$ (on the exponential) and $(1,0)$ (on the logarithm). The minimal distance is therefore the distance between those two points, which is $\sqrt2$.anemone said:What is the minimal distance between $$y=e^x$$ and $$y=\ln x$$?