If the background is white and you want it to be transparent, do this:
1. Open the file and if the layer is locked, trash the lock.
2. Cmd+click on the New Layer symbol in the Layers panel to place a blank layer below the logo layer.
3. Return to the top layer and double click on it in the Layers panel to bring up Layer Style.
4.. In the Blend If section at the bottom of Layer Style, move the right slider of the top grayscale to the left until the white in the illustration disappears.
5. Layer > Merge down and then File > Save
One more suggestion: If you are converting the pure white background to transparent so that you can then place it, as a layer, on another image, there is another possible approach,
Move the image with the white background to the target image and then change its Blending Mode to Multiply.
Job done.