Preheat the oven to 350 degrees F (175 degrees C). Grease a metal loaf pan with butter.

Whisk sugar and lemon zest together in a large bowl. Whisk in yogurt, eggs, oil, and vanilla extract until well blended.

Mix flour, baking powder, and salt together in a separate bowl. Whisk into the yogurt mixture until just blended. Pour batter into the prepared loaf pan.

Bake in the preheated oven until a toothpick inserted into the center comes out clean, about 1 hour. Cool in the pan for 10 minutes. Invert cake onto a cooling rack and cool completely.