I have struggled with this before as well and I hope this helps a little bit. I copy and pasted it from gotquestions.org.
There is no verse in the Bible that says, “Lucifer is Satan,” but an examination of several passages reveals that Lucifer can be none other than Satan. The fall of Lucifer described in Isaiah 14:12 is likely the same that Jesus referred to in Luke 10:18: "I saw Satan fall like lightning from heaven.” A similar fall is depicted in Ezekiel 28.
Isaiah 14:12–18 describes the fall from heaven of one called “Lucifer,” a name that means “morning star,” “son of the dawn,” “Day Star,” or “shining star.” The description of the one referred to shows us it can be none other than Satan. We know from Jesus’ own words in Luke 10 that Satan fell from heaven. So, when Isaiah refers to Lucifer (In Hebrew, helel) being cast down to earth (Isaiah 14:12), it can be none other than Satan. The reason for his fall is found in verses 13 and 14: “You said in your heart, ‘I will ascend to heaven; I will raise my throne above the stars of God; I will sit enthroned on the mount of assembly, on the utmost heights of the sacred mountain. I will ascend above the tops of the clouds; I will make myself like the Most High.’” This has always been Satan’s desire—to be God—and it is the very temptation he used in the garden of Eden to get Eve to disobey God: “You shall be as God” (Genesis 3:5).
Ezekiel 28 is another passage thought to refer to Lucifer/Satan. Although it begins with Ezekiel being commanded by God to “take up a lament concerning the king of Tyre” (v. 12), an evil idolatrous king, it soon becomes clear that the passage is referring as well to the power behind that king—Satan. Verse 13 says he was “in Eden, the garden of God.” Clearly, the king of Tyre was never in Eden. Verse 14 says, “You were anointed as a guardian cherub, for so I ordained you.” Apparently, Lucifer/Satan had a position of guardian angel in heaven “among the fiery stones,” thought to be the shining precious jewels that are seen in other descriptions of heaven (Exodus 24:10; Revelation 21:18–21). Since the king of Tyre was never in heaven, either, this can only be describing Lucifer. The rest of the passage describes the reason he was cast out of heaven. Because of his beauty, his heart became proud and his wisdom was corrupted (Ezekiel 28:17). Pride in his perfection, wisdom, and beauty (v. 12) became the source of his downfall, and God threw him to the earth (v. 17). This was witnessed by the Lord Jesus in heaven before His incarnation (Luke 10:18).
To summarize, Lucifer was cast out of heaven for his sin of pride and his desire to be God. Jesus referred to seeing Satan being cast out of heaven. Therefore, we can conclude that Lucifer and Satan are one and the same.