The functions f(x)= ex, x in R, and g(x)=ln x, x>0, undo one another: ln(ex)=x and eln x=x when x>0. The statement y= ex is equivalent to x=ln y. Swapping coordinates reflects a graph in y=x. The exponential range (0,infinity) becomes the logarithm domain; the horizontal asymptote y=0 becomes x=0. Method: write y in place of the function value; swap x,y; solve for y; exchange domain and range. For a restricted original domain, find its actual range before stating the inverse domain. If f(x)= ex for x 0, then f-1(x)=ln x for x 1, with range [0,infinity). Exam check: ln(ex)=x holds for all real x, whereas eln x=x requires x>0. An inverse is not the reciprocal of a function.

The functions \(\mathrm f(x)=\mathrm e^x\), \(x\in\mathbb R\), and \(\mathrm g(x)=\ln x\), \(x>0\), undo one another: \(\ln(\mathrm e^x)=x\) and \(\mathrm e^{\ln x}=x\) when \(x>0\).

The statement \(y=\mathrm e^x\) is equivalent to \(x=\ln y\). Swapping coordinates reflects a graph in \(y=x\). The exponential range \((0,\infty)\) becomes the logarithm domain; the horizontal asymptote \(y=0\) becomes \(x=0\).

Method: write \(y\) in place of the function value; swap \(x,y\); solve for \(y\); exchange domain and range. For a restricted original domain, find its actual range before stating the inverse domain.

If \(\mathrm f(x)=\mathrm e^x\) for \(x\ge0\), then \(\mathrm f^{-1}(x)=\ln x\) for \(x\ge1\), with range \([0,\infty)\).

Exam check: \(\ln(\mathrm e^x)=x\) holds for all real \(x\), whereas \(\mathrm e^{\ln x}=x\) requires \(x>0\). An inverse is not the reciprocal of a function.

Natural exponential and logarithm as inverses

Natural exponential and logarithm as inverses