solve $x=e^{a\frac{\ln(b/x)}{\ln(b/x)+c}}$ for $x$

Question

Solve the following equation for $x$: $$x=e^{a\frac{\ln(b/x)}{\ln(b/x)+c}}$$ where $a,b,c>0$

From the curve, I see it has two solutions but I cannot find the exact answer. Any idea what is the answer?

Answer 1

Hint. Since the RHS is an exponential, necessarily the LHS $x$ should be positive. Moreover the term at the denominator, $\ln(b/x)+c$, should be different from zero.

Then, taking the natural logarithm of both sides we get $$\ln(x)=a\frac{\ln(b/x)}{\ln(b/x)+c}.$$ Now recall that $\ln(b/x)=\ln(b)-\ln(x)$ (here $b>0$). Can you take it from here?