Verified answer

I would say it is what you have stated, rsinθ=e^(rcosθ)

I don't think you can express it in the form r=f(θ) if that's what you are hoping.

I agree with the first answerer. Because there is a transcendental relationship between r and θ, it does not appear that there is a simple equation. You can use the standard transformations to find a parametic relation between r, θ and x, however. It looks like:

θ = Arc tan (y/x) = Arc tan (e^x/x) and

r =√(x² + y²) = √(x² + e^(2x))