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They may turn more yellow and grow softer, but they will not necessarily become sweeter or juicier. If you get an under-ripe lemon and leave it on your counter, it may turn more yellow but it will still taste bitter. The ideal place would be on your kitchen counter, away from direct sunlight. After a few days, the lemon will turn more yellow. Keep in mind, however, that this will not actually ripen the lemon on the inside; the lemon may still taste bitter and under-ripe even though it is yellow. It will, however, still work well as a garnish if you want to add some color to a dish or drink. A greenish lemon does not necessarily mean that it is not yet ripe. You can still have a ripe lemon, even if it looks a little green. This is because lemons ripen from the inside out. The flesh ripens first, then the skin. If you find yourself with an under-ripe lemon, try cutting it open anyway and tasting it. You could also follow the steps listed in this method for determining whether or not a lemon is ripe. Instead, consider using it to clean, or as an air fresher.  You can make an effective rust and tarnish busting scrub by mixing salt and lemon juice into a paste. Add a few slices of lemon into a pot of simmering water. You can also add a few other fragrant herbs to compliment the lemon, such as rosemary.
Know that lemons will never truly ripen once you take them off the tree. Try placing an under-ripe lemon in a bright location in your kitchen. Understand that color is not a good indicator of ripeness. Don't let an under-ripe lemon go to waste.