While working with exponential growth and decay, I encountered a problem where I need to solve an equation involving logarithm. I could not separate or could not make it explicit.
$y=\frac{\ln((1+r)(1-\delta))}{\ln\bigl(\bigl(1+\frac{r}{y}\bigr)(1-\delta)\bigr)}$. where $r$ and $\delta$ are positive, both are smaller than $1$.
I need the solution of this in the form of; $y=f(r,\delta)$. I tried numerical solution also but could not find a best way. Any hint or help is appreciated.
$$y=\frac{\log ( (1+r)(1-\delta ))}{\log \left( \left(1+\frac{r}{y}\right)(1-\delta )\right)}$$ For the time being, let $$a=\log ( (1+r)(1-\delta ))\qquad \text{and} \qquad b=\log(1-\delta)$$ which make $$y=\frac a {\log \left(1+\frac{r}{y}\right)+b}\implies \frac 1y=\frac {\log \left(1+\frac{r}{y}\right)+b} a$$ Now, some basic manipulations $$\left(1+\frac{r}{y}\right)=\left(1+\frac{b\, r}{a}\right)+\frac r a\, \log \left(1+\frac{r}{y}\right)$$ Let $$c=1+\frac{b\, r}{a} \qquad \text{and} \qquad d=\frac r a \qquad \text{and} \qquad x=1+\frac{r}{y}$$ which make $$x=c +d \,\log(x)\implies x=-d\, W\left(-\frac{1}{d}\,e^{-\frac{c}{d}}\right)$$ where appears Lambert function.
Go back to $y$ and replace $(a,b,c,d)$ by their definitions in terms of $(r,\delta)$
