Adjust the oven rack to the middle position and preheat the oven to 350ºF (180ºC). Grease and flour an 8 1/2-x-4 1/2-inch (23.5-x-12.7-cm) loaf pan. You can also use a nonstick baking spray with flour.
Whisk the flour, baking powder, baking soda, cinnamon, ginger, nutmeg, and salt together in a large bowl until combined.
In another bowl, mix melted butter, brown sugar, maple syrup, applesauce, yogurt, and vanilla extract. Mix until well combined.
Add wet ingredients mixture into dry ingredients; mix until just combined.
Fold the grated carrot into the batter thoroughly.
Pour the mixture into the prepared loaf pan.
Bake for 50 to 55 minutes, or until a toothpick inserted in the center comes out clean.
Cool in the pan on a wire rack for 15 minutes, then remove the loaf from the pan and set on the wire rack to cool completely.
Serve warm or at room temperature.