Preheat the oven to 350 degrees F (175 degrees C). Grease a 9-inch round springform cake pan.

Empty the yogurt into a mixing bowl. Measure the oil, using the yogurt container, and add to the yogurt. Set aside. Rinse yogurt container and dry with paper towel to continue using it as a measuring tool.

Add eggs, sugar, and vanilla extract to the yogurt-oil mixture. Beat well until thoroughly combined. Add flour, baking powder, and salt. Stir until just combined; batter will be thick. Pour into the prepared cake pan and spread to level the top.

Bake in the preheated oven until a toothpick inserted in the center comes out clean, 25 to 30 minutes.