Inverse function to $f(t)=3t+4ln(t+1)=y$

Question

I have to invert the function $f(t)=3t+4\ln(t+1)=y$, so $f^{-1}(y)=t$. But I am struggling to invert this. Is there a solution?

Answer 1

The Lambert W function $W(y)=z$ has the property that if $y=ze^z$, then $z=W(y)$.

So the standard approach is to try to get the expression into the form $Y=Te^T$, where $T$ is something from which we can easily get $t$ and $Y$ is a function of $y$.

So in this case we start by exponentiating both sides: $e^{\frac{y}{4}}=(1+t)e^{\frac{3}{4}t}$. Multiply both sides by $\frac{3}{4}e^{\frac{3}{4}}$ we get $\frac{3}{4}e^{\frac{y+3}{4}}=\frac{3}{4}(1+t)e^{\frac{3}{4}(1+t)}$

Hence if we set $Y=\frac{3}{4}e^{\frac{y+3}{4}}$ and $T=\frac{3}{4}(t+1)$ the equation becomes $Y=Te^T$. So we can use the property of the Lambert function to get $T=W(Y)$, so $t=-1+\frac{4}{3}W(\frac{3}{4}e^{\frac{y+3}{4}})$.