The doubling rumor
A rumor doubles each hour — exponential growth. Asking 'how many hours until a thousand people know?' is a logarithm: the question mark that runs the doubling backward.
Inverse operations
Exponent laws: aᵐ·aⁿ = aᵐ⁺ⁿ. A logarithm answers 'what power?': log_a(aⁿ) = n.