Exponential Growth and Decay (F 702)

A rumor that doubles every hour starts slow but quickly outruns one that adds ten people every hour, because doubling multiplies the current total while adding only tacks on a fixed chunk. After enough hours the multiplied pile dwarfs the added pile, since multiplying by 2 repeatedly grows faster than any fixed amount added repeatedly. This is the core difference between exponential change and linear change: one scales with the size you already have, the other does not care how big you are.

A growth factor packages a percent change into one number you multiply by. Growing 7 percent means you keep the original 100 percent and add 7 more, so you multiply by 1.07. Losing 15 percent means you keep 85 percent of what you had, so you multiply by 0.85. Apply that factor once per time period, and after n periods you have multiplied by it n times: value = start x (factor)^n. Watch the exponent especially closely when the rate is given per year but the problem counts in months, or per some chunk of minutes but asks for hours — convert the clock first, then build the exponent from the number of periods, not the raw time elapsed.

This connects directly to logarithms: once you can build the equation start x (factor)ⁿ = target, finding n means solving for a variable trapped in an exponent, which is exactly what logs are for. The growth-and-decay setup is the reason logs exist on this test — every log law and every log-solving technique traces back to needing to undo an exponent like the one here.