A logarithm answers the question “what power do I raise this base to, to get this number?” log₁₀(1000) = 3 because 10³ = 1000. It is exactly the inverse of exponentiation.
The three bases you actually meet
Base 10 for orders of magnitude, decibels, pH and the Richter scale. Base e ≈ 2.71828 for anything involving continuous growth or calculus, because the derivative of eˣ is itself. Base 2 for computer science, where it counts bits and the depth of binary trees.
The properties that make logs useful
Logarithms turn multiplication into addition. Before calculators this is why slide rules and log tables existed, and it remains why logarithmic axes make exponential data legible: a curve that doubles at a constant rate becomes a straight line.
Change of base
Most calculators only have ln and log₁₀, but any base can be obtained by dividing: log₇(50) = ln(50)/ln(7). It does not matter which of the two you use as long as you use the same one twice.