Key Idea: Logs undo powers. You use them to combine and split log terms, to change the base, and to solve equations where the unknown is a power, such as a half-life question.
Log laws
- × inside → + logs; ÷ → −; power → × (same base throughout).
- Change base: logₐx = log x / log a (or use ln).
- aˣ = b → x = log b / log a. Taking logs drops the unknown power down so you can divide it out.
- Rewrite logₐ N = k as aᵏ = N. Then check that what sits inside each log stays > 0.
Half-life: when does A = 80(0.5)ᵗ/⁶ reach 10 mg?
- Divide by 80 — (0.5)ᵗ/⁶ = 1/8
- Spot the power — 1/8 = (0.5)³, so t/6 = 3
- Answer — t = 18 h, which is three half-lives
A log turns a hidden power into a number you can divide by.
Is log x + log y the same as log(xy)? Yes. A product inside one log splits into a sum of two logs.
How do you find log₂ 7 on a calculator with no base 2 key? Work out log 7 ÷ log 2.
Your answer to a log equation makes a log's inside negative. What now? Reject it. Logs only take positive inputs.
A sample halves every 6 hours. What fraction is left after 18 hours? 1/8, after three halvings.