NapkinCalc

Algebra 2B — Exponentials, Logs & Sequences

Logarithms — the inverse question

continues from lesson 1 — values defined earlier in the course stay live here

ELI5: a logarithm asks "what power?" — log_b(y) is the exponent you must raise b to in order to get y. It is the exact undo of exponentiation. And it turns multiplication into addition: log(a·b) = log a + log b (the trick that powered slide rules).

pdouble=log(2)log(1+0.07)p_{double} = \frac{\mathrm{log}\left(2\right)}{\mathrm{log}\left(1 + 0.07\right)} = 10.244810.2448 periods to double at 7 % — the honest "rule of 72"
✓ pass abs(1.07pdouble−2)<0.000001\mathrm{abs}\left(1.07^{p_{double}} - 2\right) < 0.000001 raising 1.07 to that power really gives 2
✓ pass abs(log(3⋅5)−log(3)+log(5))<10−9\mathrm{abs}\left(\mathrm{log}\left(3 \cdot 5\right) - \mathrm{log}\left(3\right) + \mathrm{log}\left(5\right)\right) < 10^{-9} the log law: log(3·5) = log 3 + log 5

Real-world hook: logs measure things that span huge ranges — decibels (sound), pH (acidity), the Richter scale (earthquakes), and "orders of magnitude" everywhere.

Try it yourself: solve 2ˣ = 32. (What power of 2 gives 32?)

xyou=5x_{you} = 5 ✏️ Your turn: find x with 2ˣ = 32. The check raises 2 to your x and compares to 32.
✓ pass abs(2xyou−32)<0.000001\mathrm{abs}\left(2^{x_{you}} - 32\right) < 0.000001 green when 2ˣ equals 32