Logarithms continue to play an important role in mathematics (most significantly in calculus), science, and engineering. It is therefore important for students to understand logarithms as real numbers as well as the characteristics of logarithmic functions. Exploration of challenges in understanding logarithms as real numbers and logarithmic functions as well as their graphs provides insight that can be used as the basis for instruction. This paper discusses and shares evidence of students' difficulties collected from various courses over time. The authors share concepts related to logarithms that can help students build an understanding of these functions, and present some ways that misconceptions related to these concepts are manifested to suggest what teachers can listen for as they explore logarithms with students.