Preheat oven to 350°F. Grease and flour a round pan or standard loaf pan. In a medium bowl, whisk together 1 1/2 cups all-purpose flour, 2 teaspoons baking powder, and 1/2 teaspoon salt.
Using your fingers, mix the sugar with the lemon zest in a large bowl until the sugar has the texture of damp sand. Add the yogurt and lemon juice, and mix well. Then add the vegetable oil, eggs, vanilla, and basil. Whisk until well-blended.
Carefully fold in the dry mixture until just combined. Pour into prepared pan. Bake 40 to 50 minutes (adjust time according to the type of pan you used), or until a toothpick comes out clean.