Recreate Logo in High Resolution

I have a low-resolution reference of my logo that now needs to look sharp and professional online. The end product I’m after is a clean, high-resolution PNG sized for web use, with a transparent background and crisp edges that hold up on retina displays... (Budget: $30 - $250 AUD, Jobs: Adobe Illustrator, Adobe Photoshop, Digital Art, Graphic Design, Illustrator, Image Processing, Logo Design, Photoshop, Vector Design, Visual Design)