# Solving a transcendental equation of the form : $\log (y)-a y=b$

I am solving a physical problem, where I try to solve analytically a transcendental equation of type:

$$\log (y)-a y=b,$$

here $a$ and $b$ are constants.

The problem is that I get a solution but with an error message who indicate that some solutions may not be found:

Solve[Log[y] - a y == b, y]

(* {{y -> -(ProductLog[-a E^b]/a)}} *)
(* Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information. *)


Is this the right solution? Please how to found it?

• Note that in Mathematica Log[y] is a natural logarithm of y. A logarithm with base 10 is Log10[y] or Log[10,y] Jul 7, 2018 at 0:07
• yes, I know it @roman465. Thank you. Jul 7, 2018 at 0:12
• If you look at the result of Simplify[Reduce[Log[y] - a y == b, y]] then you might learn a little more about this. If you know that your constants are Real then you can let Reduce and Simplify know that and they might do an even better job of helping you understand the results.
– Bill
Jul 7, 2018 at 1:05
• If you plot Log[y] and a+b y you can see that, depanding on a,b, there exist no, one or two intersections (roots of the equation). Jul 7, 2018 at 12:53

ProductLog[z]

gives the principal solution for w in z==we^w. After all use
Solve[Log[y] - a y == b, y] // Quiet

• @Gallagher, to get all solutions, you need to use the other branches of ProductLog[]; look up the documentation for this. Apr 17, 2020 at 13:40