Is there a good way to solve for z the equation $e^{i\pi} = e^{z\ln2} + e^{z\ln3}$?

$e^{i\pi} = e^{z\ln2} + e^{z\ln3}$

How can I deal with this? I want to solve for z. Does this help?

$e^{z\ln2} + e^{z\ln3} = e^{z\ln2}(1 + e^{z(ln3-ln2)})$

If I write out z=x+iy then the expression becomes

$-1 = e^{x\ln2}e^{iy\ln2}+e^{x\ln3}e^{iy\ln3}$

To the best of my knowledge, you can't solve for $z$ analytically. As in my comment above, the expression simplifies to $-1=2^z + 3^z$, but there isn't a way to find $z$ explicitly. Even separating into real and imaginary parts yields a system of nonlinear equations. So to answer: is it solvable? No, it is not. So you'd have to use numerical methods to solve $0=2^z + 3^z -1$. But in terms of interpreting it, I hope someone can provide more insight.

Write $log(2)=A+2\pi i n$ and $log(3)=B + 2\pi i m$ where $A$ and $B$ are the real logs of $2$ and $3$.

Write $z=x+iy$, write out the real and imaginary parts of your equation and you'll get two equations in the four unknowns (two real, two integers) $x,y,m,n$, and it won't be hard to characterize the solutions.