Mathematics from zero
Growth: the race and the ruler
Reading that exponential growth wins is not the same as watching it happen in your own numbers. Run a race between adding and multiplying day by day, find the exact day the multiplier pulls ahead, and then turn it around — use a logarithm to count how many doublings it took. One table teaches the whole unit.
Turn the unit’s ideas into something you can see and check: build a linear column and an exponential column side by side, locate the crossover, then read totals backward with a logarithm and confirm each answer by running the exponent forward.
Build a day-by-day growth comparison table by hand (paper or a spreadsheet), watch exponential growth overtake linear growth, find the exact crossover day, and then use a base-2 logarithm to count the doublings behind several totals — checking every log by running the exponent forward.
- A completed days 0–12 table with both columns correct, the early days where linear leads marked, and the crossover day stated.
- A correct base-2 logarithm next to each power-of-2 total, each verified by writing out 2 to that power and confirming it matches.
- A short written explanation of why exponential overtakes linear, framed as fixed step versus growing step — not just 'it gets bigger.'
- Correct values for log₁₀(10), log₁₀(100), and log₁₀(1000), with a one-line note tying the base-10 log of a round number to its count of zeros.
- Add a third column M that starts at 1 and triples each day. Predict before filling it in whether M overtakes E sooner or later than E overtook L, then check, and explain the result in terms of the multiplier.
- Plot L and E on the same axes (graph paper or a chart) and mark the crossover point visually — see the linear straight line and the exponential curve bending upward.
- Estimate log₂(100) to one decimal place by reasoning that it is between 6 and 7 and closer to 7 (since 100 is much nearer 128 than 64), then compare against a calculator's value.
- Find a real doubling story — a savings account, a population, or a piece of news spreading — write its starting value and multiplier, and use a base-2 logarithm to estimate how many doublings it would take to reach a target you choose.
This is the whole unit in one worksheet: a linear column that adds, an exponential column that multiplies and eventually races past it, and a logarithm that runs the exponential backward to count the doublings. Once you have built the table, marked the crossover, and checked each log by running its exponent forward, the relationship between exponential growth and the logarithm stops being two definitions and becomes one picture you can rebuild from memory.