Understand a logarithm as the inverse of an exponential, graph logarithmic functions, and evaluate common and natural logs.
A logarithm answers the question an exponential equation leaves hanging: log_b(x) is the power you must raise b to in order to get x. Because logarithms and exponentials undo each other, log_b(b^x)=x and b^(log_b(x))=x for any valid x -- they are inverse functions of one another, and their graphs are mirror images across the line y=x.
Two bases are common enough to get their own shorthand: log(x) with no written base means base 10 (the "common log"), and ln(x) means base e (the "natural log"). A logarithmic graph passes through (1,0), has a vertical asymptote at x=0, and its domain is x>0 only -- a logarithm can never accept 0 or a negative input, since no power of a positive base ever produces one.
Mastery looks like: They evaluate common and natural logarithms fluently, solve simple logarithmic equations by converting to exponential form, and state the domain of a logarithmic function.
Common stumbles: Forgetting that a logarithm's input must be positive when stating a domain; mixing up which number is the base and which is the answer.
See it visually
Interactive, spoken-aloud tools that show this idea in motion.