Equivalent Forms and Percent Change

An exponential function grows by the same factor over equal intervals, but that interval is your choice, not a fact about the quantity. A population doubling every week is also growing by some smaller percent every day, because a week is just 7 days glued together. Rewriting the function changes the unit you're measuring growth in, not the growth itself.

The algebra behind this is exponent rules. If a quantity doubles every 7 days, its value after d days is 2^d/7. Rewrite this as (2^1/7)^d and you now have a daily growth factor: r = 2^1/7, about 1.1041. Since r is the amount you multiply by each day, the daily percent increase is (r - 1) times 100, about 10.41%. The same quantity, same graph, just described one day at a time instead of one week at a time.

This matters because every exponential model y = a·b^x hides a percent change in its base: b is always 1 plus the decimal rate per step. A base of 1.05 means 5% growth per step; a base of 0.92 means 8% decay per step. Once you can move freely between 'doubles every 3 days' and 'grows 26% daily' and 'base of 1.26,' you're ready to match real data to a model, build models from word problems about percent change, and eventually compare exponential growth to other function families.