Two symbols show up all over growth problems: the number e and the button ln on your calculator. Here is what they are, where they come from, and how ln undoes e.
Interest is added a set number of times per year (yearly, monthly, daily).
A = P(1 + r⁄n)nt
Example: $1000 in a savings account at 6%, compounded monthly, for 1 year:
1000(1 + 0.06/12)12·1 ≈ $1061.68
No steps at all — the amount is always growing. This is where e appears.
A = Pert
Example: 200 bacteria grow continuously at 5% per hour. After 10 hours:
200e0.05·10 = 200e0.5 ≈ 330 bacteria
Take the same $1000 at 6% for 1 year, and compound it more and more often:
| Compounded | n | Amount after 1 year |
|---|---|---|
| Yearly | 1 | $1060.00 |
| Quarterly | 4 | $1061.36 |
| Monthly | 12 | $1061.68 |
| Daily | 365 | $1061.83 |
| Continuously (1000e0.06) | ∞ | $1061.84 |
The more often you compound, the closer you get to the continuous answer — but it never runs away to infinity. It levels off. That ceiling is set by e. Here's the pattern with 100% growth:
| n (times per year) | (1 + 1/n)n |
|---|---|
| 1 | 2 |
| 12 | 2.61304 |
| 365 | 2.71457 |
| 1,000,000 | 2.71828 |
| ∞ | e = 2.71828… |
Related: continuous growth is what compound growth turns into when you compound infinitely often.
Different: compound growth jumps at set times and uses (1 + r/n)nt; continuous growth never stops and uses ert. Continuous always gives slightly more.
| Log base 3 | because… | ln (log base e) | because… |
|---|---|---|---|
| log3 9 = 2 | 32 = 9 | ln e2 = 2 | e2 = e2 |
| log3 81 = 4 | 34 = 81 | ln e5 = 5 | e5 = e5 |
| log3 3 = 1 | 31 = 3 | ln e = 1 | e1 = e |
| log3 1 = 0 | 30 = 1 | ln 1 = 0 | e0 = 1 |
| log3 ⅓ = −1 | 3−1 = ⅓ | ln ¹⁄ₑ = −1 | e−1 = ¹⁄ₑ |
With "messy" numbers, use the ln button on your calculator:
Just like subtracting undoes adding and square roots undo squaring, ln and e undo each other:
ln(ex) = x eln x = x
Take ln of both sides, and the e disappears — the exponent drops down where you can solve for it.
Always: get the e part alone first, then take ln.
Use ln to undo e. Round to three decimal places. Try each one before opening the answer.