Additional Mathematics – Logarithmic and exponential functions | e-Consult

Two functions are inverses when swapping the variables in one yields the other. Starting with \(y=e^{x}\) and interchanging \(x\) and \(y\) gives \(x=e^{y}\). Solving for \(y\) gives \(y=\ln x\); hence the two functions are inverses.

The graph of a function and its inverse are reflections in the line \(y=x\). Any intersection of the two graphs must therefore satisfy \(y=x\). Substituting \(y=x\) into \(y=e^{x}\) gives the equation

\(e^{x}=x\).

For real \(x\), \(e^{x}>x\) for all \(x\) (since the exponential curve lies above the line \(y=x\) and they are tangent only at a complex point). Consequently the equation has no real solution.

Therefore the graphs of \(y=e^{x}\) and \(y=\ln x\) do not intersect at any real point.