The folded-paper fortune
A rumor doubles each hour: 1, 2, 4, 8 — that runaway growth is exponential. The logarithm runs the film backward, answering 'how many hours until a thousand people know?' It's the question-mark that undoes exponential growth.
Repeated multiplication and its undo
An exponential function b^x multiplies by the base b each time x increases by one, producing rapid growth (b > 1) or decay (0 < b < 1). The logarithm is its inverse: log_b(y) answers 'to what power must b be raised to get y?' So b^x = y is the same statement as log_b(y) = x.
Logs turn products into sums
Because exponents add when powers multiply, log(mn) = log(m) + log(n). This identity is why logarithms tamed hard multiplication before calculators.
2^3 = 8, so log₂(8) = 3. And log₂(4·8) = log₂4 + log₂8 = 2 + 3 = 5, matching log₂(32).
Lab · Undo the exponent
- You know that 10^x = 1000.
- Rewrite the statement as a logarithm.
- Evaluate to find x.
What you should see: 10^x = 1000 means log₁₀(1000) = x, and since 10³ = 1000, x = 3.