Every growth-and-decay problem is the same equation — an unknown trapped in an exponent — asked from a different angle. Drag the curve to solve for the time and watch the logarithm do the work live, then step through the three questions the model can answer: how much, how long, and how fast.
Drag until the curve hits your target
Starting amount A0100
Doubling time T (hours)8
Target (× the start)3.00×
Why a logarithm is the tool. The unknown t lives upstairs, in the exponent, so adding and dividing can never reach it. A logarithm is the one operation that pulls an exponent down to the ground: log(bx) = x·log b. Once t is a plain coefficient the equation is linear, and linear equations we already know how to finish.
Three questions, one modelpick a question below
What are we solving for?
Read the question before you reach for the log. If the ratio of new to old is a tidy power — 2, 4, 8, ½, ¼ — both sides share a base and matching exponents finishes it with no calculator. Only when the ratio is untidy (0.30, 0.62, 1.07…) does the logarithm earn its keep.