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In a large bowl, combine the sugars, applesauce, and egg.
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Add in the zucchini, vanilla, cinnamon and salt.
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Finally add in the flour and baking soda. Combine until no lumps of dry ingredients remain.
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Grease an 8x8 baking dish and pour in the mixture. Bake at 350 degrees for 35-40 minutes or until a toothpick inserted into the center of the cake comes out clean.
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This cake can be doubled and placed in a 9x13 pan. Cool slightly before serving. You could also top it with some Cream Cheese Frosting for a special treat!