Exponential functions f(x) = a·bˣ — the starting value, the growth factor, and how a percent becomes a factor; growth and decay; linear versus exponential in a table, a graph and a story; doubling time and half-life; compound and continuous interest. Then the logarithm as the inverse question — 2 to what power is 5? — with the product, quotient and power rules, change of base, and the routine that gets a variable out of an exponent: 500 = 100·2ᵗ gives t = log 5 / log 2 ≈ 2.32. Study cards, hints, a practice quiz at three levels, and four workshop activities that check themselves.
Start with an exponent you already know how to read: 2 to the 5th is 32.
A log is an exponent asking to be found. log_b(y) = x and b^x = y are the same sentence read from two ends — so anything you can do to one, you can do to the other.
What people get wrong
⚠️People often think…
3·2⁴ means 6⁴.
The exponent belongs to the 2 alone. Do the power first: 2⁴ = 16, then 3 × 16 = 48. The starting value waits its turn.
Power first, then the starting value.
⚠️People often think…
5% growth means multiplying by 0.05 each year.
The factor is what is LEFT, not what changed. 5% growth is ×1.05; 12% decay is ×0.88. Multiplying by 0.05 would wipe out 95% of it.
Growth: 1 + r. Decay: 1 − r.
⚠️People often think…
log(x + y) = log x + log y.
Only PRODUCTS split into sums. log₂ 8 + log₂ 4 = log₂ 32 = 5, not log₂ 12. Sums inside a log do not come apart at all.
Products split. Sums do not.
⚠️People often think…
To solve 100·2ˣ = 500, take the log of both sides right away.
Isolate the power first. Divide by 100: 2ˣ = 5, then x = log 5 / log 2 ≈ 2.32. Logging too early solves a different equation.
Isolate the power, then take the log.
Isolate, log, drop, divide
1Watch one
Solve 100·2ˣ = 500.
Isolate the power — divide both sides by 100.
2ˣ = 5.
Take the log of both sides; the exponent drops in front.
x = log 5 / log 2 ≈ 2.32. Check: 2²·³² ≈ 5. ✓
2Do one with me
A $400 account grows 5% a year. Fill in the model.
The starting value is
The growth factor is
log₂ 8 + log₂ 4 =
💬One sentence, then you move on
Why do you isolate the power before taking a log?
3Try one
Write log₂ 5 using common logs (change of base).
I want a hint first
2² = 4 and 2³ = 8, so the answer sits between 2 and 3. Only one of the two fractions does.
💬Last one — then you're done here
Why can a log equation produce an answer you have to throw out?
Where this goes
Where this lives
Interest on a balance, a medicine leaving the blood, a rumor spreading. Each multiplies by the same factor every step — and the log is how you ask how long it takes.
What this feeds
Next unit is trigonometric functions, where the pattern repeats instead of climbing.
Name one thing you know of that grows or shrinks by a percent each time.
One card at a time — tap “Show me” to check yourself, then Next. Start at Foundation; when those feel easy, climb.
Helpful Hints
🔑 Solving an exponential equation — the routine
ISOLATE the power (divide away the starting value: 500 = 100·2ᵗ → 2ᵗ = 5) → ESTIMATE (2² = 4, 2³ = 8, so t is between 2 and 3) → take the LOG of both sides (any base; ln for e) → POWER RULE brings the exponent down (t·log 2 = log 5) → DIVIDE (t = log 5 / log 2 ≈ 2.32) → CHECK by plugging back in (100·2^2.32 ≈ 499).
Isolate, estimate, log, drop, divide, check.
🔁 The whole topic in one idea
An exponential multiplies by the same factor every step; a logarithm asks “how many steps?” log_b(y) = x means exactly bˣ = y — a log IS an exponent. That is why log rules are exponent rules in disguise (products add, powers multiply) and why the only way to get a variable out of an exponent is to take a log.
A log is an exponent asking to be found.
⚠️ Traps the test loves
Multiplying the starting value by the base first: 3·2⁴ is 3·16 = 48, not 6⁴. The exponent belongs to the base alone.
Using the percent as the factor. 5% growth is ×1.05; 12% decay is ×0.88 — what is LEFT, not what is lost.
Taking logs before isolating the power: log 500 / log 2 ≈ 8.97 solves the wrong equation. Divide by 100 first.
Inventing log(x + y) = log x + log y. Only PRODUCTS split: log₂ 8 + log₂ 4 = log₂ 32 = 5, not log₂ 12.
Flipping change of base: log₂ 5 = log 5 / log 2 ≈ 2.32. Argument on top, base on the bottom; log 2 / log 5 ≈ 0.43 is log₅ 2.
Keeping a solution that makes a log of a negative number. log x + log(x − 3) = 1 gives x = 5 or −2; only 5 survives.
A table or story — “linear or exponential?” Subtract neighbors for linear; divide neighbors for exponential; look for “adds” versus “percent” or “doubles.”
“Write the function” from a rate and a start, or from two points (a from x = 0, b from the ratio).
“Evaluate” log₂ 32, log 0.001, ln e², log₃(1/9) — ask “base to what power?”
“Solve for t” — a common base if you can (9ˣ = 27), a logarithm if you cannot (2ᵗ = 5), ln when the base is e.
“Expand” or “condense” a log expression using the product, quotient and power rules.
A log equation — rewrite as an exponential, solve, then reject anything that makes a log argument ≤ 0.
A graph — the asymptote (y = k for bˣ + k; x = 0 for log x), the intercept ((0, 1) or (1, 0)), growth or decay from the base.
✅ Before the test, can you…
Turn “grows 5% a year” and “loses 12% a year” into factors 1.05 and 0.88, and evaluate a·bˣ without multiplying a by b first?
Tell linear from exponential in a table, a graph and a story?
Evaluate log₂ 32, log 1000, ln e and log₃(1/9) in your head?
Solve 500 = 100·2ᵗ and get t = log 5 / log 2 ≈ 2.32, then check it?
Use the product, quotient and power rules — and explain why there is no rule for log(x + y)?
Find the doubling time at 7% two ways (rule of 70 and ln 2 / ln 1.07) and say how far apart they are?
Write the inverse of f(x) = 2^(x+1) and check it with one point?
Pick your level
Look back at anything you missed — the hint that appeared is exactly what to reread tonight.
How sure did you feel?
Workshop
Work like a mathematician: graph a colony that doubles every hour and watch the bars race away, put the solution of 500 = 100·2ᵗ in order, sort twelve tables and stories into linear, growth and decay, then argue whether the rule of 70 is good enough for a town planner. Every activity checks itself, and hints are free.
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Your practice record — saved on this device
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