Exponential and logarithmic inverse pairs

\(y=\mathrm{e}^{x}\) and \(y=\ln x\) are an inverse pair: \(\ln\left(\mathrm{e}^{x}\right)=x\) for every real \(x\), and \(\mathrm{e}^{\ln x}=x\) for \(x>0\). Their graphs are reflections of each other in \(y=x\).

Reflection in \(y=x\) swaps coordinates. The domain and range swap, a horizontal asymptote \(y=c\) of \(\mathrm{f}\) becomes the vertical asymptote \(x=c\) of \(\mathrm{f}^{-1}\), and an intercept \((0,b)\) becomes \((b,0)\).

Inverse of \(y=p+q\mathrm{e}^{rx}\)

For \(\mathrm{f}(x)=1+2\mathrm{e}^{x}\), \(x\in\mathbb{R}\): \(2\mathrm{e}^{x}>0\), so the range is \(\mathrm{f}(x)>1\) and the asymptote is \(y=1\).

\(\begin{aligned}[t] y&=1+2\mathrm{e}^{x} \\ \mathrm{e}^{x}&=\dfrac{y-1}{2} \\ x&=\ln\left(\dfrac{y-1}{2}\right) \end{aligned}\)

\(\mathrm{f}^{-1}(x)=\ln\left(\dfrac{x-1}{2}\right)\), for \(x>1\). Its range is \(\mathbb{R}\), its asymptote is \(x=1\), and the intercept \((0,3)\) of \(\mathrm{f}\) becomes \((3,0)\).

Inverse of \(y=k\ln(bx+c)\)

For \(\mathrm{g}(x)=3\ln(2x+4)\), for \(x>a\): the argument must be positive, \(2x+4>0\), so \(x>-2\). The least value of \(a\) is \(-2\), and \(x=-2\) is the vertical asymptote. The range of \(\mathrm{g}\) is \(\mathbb{R}\).

\(\begin{aligned}[t] y&=3\ln(2x+4) \\ \mathrm{e}^{\frac{y}{3}}&=2x+4 \\ x&=\dfrac{\mathrm{e}^{\frac{y}{3}}-4}{2} \end{aligned}\)

\(\mathrm{g}^{-1}(x)=\dfrac{\mathrm{e}^{\frac{x}{3}}-4}{2}\), for \(x\in\mathbb{R}\). Its range is \(\mathrm{g}^{-1}(x)>-2\), and its asymptote is \(y=-2\).