Sigh. Sorry. I typed too fast there, and I answered incorrectly. Not sure why solve does not get this.

Multiply by log(u) + 1. Valid as long as u is not 1/e.

u = A*(log(u) + 1)

Transform this using x = log(u) + 1. Then u = exp(x - 1) = exp(x)/exp(1). Our problem is now:

exp(x)/exp(1) = A*x

or

exp(x) = exp(1)*A*x

Solve seems to see how to do that.

syms x A

xsol = solve(exp(x) == exp(1)*A*x,x)

xsol =

-lambertw(0, -1125899906842624/(3060513257434037*A))

usol = exp(xsol - 1)

usol =

exp(- lambertw(0, -1125899906842624/(3060513257434037*A)) - 1)

Verify this satisfies the original problem.

vpa(simplify(subs(u/(log(u)+1)-A,u,usol)))

ans =

0.00000000000000011018891328384950189261640307115*A

It looks like solve used a floating point approximation for exp(1) in there, so we still got zero, plus some floating point trash.

## Best Answer