Calculus I › Exponential & log laws › full formula sheet
Say it: “the natural log of a over b equals the natural log of a minus the natural log of b”
Log of a quotient
Division inside a log becomes subtraction outside — the partner of the product law.
Notation on this page: ln is the natural log (base e); a and b are positive real numbers.
Before this lesson: Log of a product
Where it comes from
Ratios inside logs — ln(1/x), ln(x/2), ln((x+1)/x) — are awkward until you split them. The lazy guess mirrors the product law's trap:
Kill it with a = b = e:
And kill the companion trap — the difference version of the sum-splitting error:
Here is the intuition. Division is multiplication in disguise: a/b = a·b−1. So the product law already tells us ln(a/b) = ln a + ln(b−1) — and a negative exponent flips to a minus sign (preview of the next law: ln(b−1) = −ln b). Division becomes subtraction because it was multiplication all along:
It is also the mirror of ex/ey = ex−y: ex turns subtraction into division, so ln turns division back into subtraction. The derivation below proves it from scratch.
Before reading on: a/b = a · b−1. If you already believe the log of a product splits, what must ln(a/b) become?
Derivation
Let a, b > 0. Same opening as the product law — write each as a power of e — plus one extra fact: ex/ey = ex−y.
Domain note. Needs a, b > 0. ln((−6)/(−2)) = ln 3 is fine, but it cannot split into ln(−6) − ln(−2). Safe general form: ln|a/b| = ln|a| − ln|b|. Also note the order matters: ln(b/a) = −ln(a/b) — flipping the fraction flips the sign.
Before reading on: predict ln(1/x) in terms of ln x — then check whether any constant term survives differentiation.
How to use it
The procedure, every time:
- Confirm it is a log of a quotient. ln(x/4) ✓. ln(x−4) ✗ — a difference, never splits. (ln x)/4 ✗ — the division is outside the log.
- Check positivity of top and bottom (or use |·|).
- Split: ln(x/4) = ln x − ln 4. Mind the order — top minus bottom.
- Use it before calculus. d/dx [ln(1/x)] becomes d/dx [−ln x] = −1/x — no chain rule, no quotient rule.
The one you'll use most
Read it backwards: combining
ln a − ln b = ln(a/b) — the direction limits love: limx→∞ [ln(x+1) − ln x] = limx→∞ ln(1 + 1/x) = ln 1 = 0. A difference of logs that looks divergent collapses to a single harmless log.
Split vs. chain rule
d/dx [ln(x/5)]: split first → d/dx [ln x − ln 5] = 1/x. Chain rule directly: (5/x)·(1/5) = 1/x — same answer, and the 5s that cancel are exactly the bookkeeping splitting skips.
Worked examples
Four problems, easiest first. In each one, read every step — the why of each move is the lesson.
Example 1 — the basic move: ln(12/4)
- Confirm: log of a quotient, both positive. ✓
- Split: ln 12 − ln 4. Or simplify first: 12/4 = 3, so = ln 3.
- Check. ln 12 − ln 4 ≈ 2.485 − 1.386 = 1.099; ln 3 ≈ 1.099. ✓
Your turn: Simplify ln(20/5).
Answer: ln 4.
Simplify first: 20/5 = 4. Or split: ln 20 − ln 5 ≈ 2.996 − 1.609 = 1.386 = ln 4 ✓.
Example 2 — the reciprocal: ln(1/e³)
- Split: ln(1/e³) = ln 1 − ln(e³).
- Simplify each: ln 1 = 0, and ln(e³) = 3 (inverse pair). So = 0 − 3 = −3.
- Check. 1/e³ = e−3, and ln(e−3) = −3. ✓
Your turn: Simplify ln(1/e2).
Answer: −2.
ln 1 − ln(e2) = 0 − 2 = −2 (inverse pair) ✓.
Example 3 — differentiate smart: d/dx [ln(2/x)]
- Split first: ln(2/x) = ln 2 − ln x.
- Differentiate: d/dx [ln 2 − ln x] = 0 − 1/x = −1/x. (The constant dies; only −ln x survives.)
- Check via chain rule. (x/2)·(−2/x²) = −1/x. ✓ Same — but the split path has no fractions-of-fractions.
Your turn: Differentiate d/dx [ln(5/x)], x > 0.
Answer: −1/x.
Split first: ln 5 − ln x — the constant dies, leaving 0 − 1/x ✓.
Example 4 — solving: ln(3x) − ln 3 = ln 4
- Combine the left side. ln(3x) − ln 3 = ln(3x/3) = ln x.
- Drop the logs. ln x = ln 4 ⇒ x = 4 (ln is one-to-one).
- Check the domain. x = 4 > 0, and 3x = 12 > 0. ✓: ln 12 − ln 3 = ln 4. ✓
Your turn: Solve ln(2x) − ln 2 = ln 5.
Answer: x = 5.
Combine: ln(2x/2) = ln x = ln 5, so x = 5. Domain check: 2x = 10 > 0 ✓.
Memorization tips
- Say it aloud: “log turns divide into minus.” Pair it with the product law: × → +, ÷ → −.
- Self-check: ln(e/e) = ln 1 = 0 = 1 − 1. If your rule doesn't give 0 here, it's wrong.
- Memorize the reciprocal form cold: ln(1/x) = −ln x. It appears in every other problem in this chapter.
- Order matters: ln(a/b) = −ln(b/a). Flipping the fraction flips the sign — say “top minus bottom” as you write it.
- The difference guard: ln(a−b) NEVER splits. Test with 5−3: ln 2 ≈ 0.69 vs ln 5 − ln 3 ≈ 0.51.
- Negative logs are fine: ln(1/2) ≈ −0.693. A negative log just means the argument is below 1 — don't “fix” it.
Final challenge
Five mixed questions — basics, applications, and the traps, all in one. Score 5/5 and the log of a quotient is yours.
← Back to the Calculus I formula sheet
How to learn a formula here
- Read each section in order — every section ends with a short quiz. Take it before moving on; the questions test exactly what you just read.
- Work the examples with the answers covered, then uncover one step at a time and compare.
- Finish with the final challenge — five mixed questions including the classic traps.
- Retake what you miss — every quiz reshuffles each attempt, and every answer explains itself.
Frequently asked questions
What is the log-of-a-quotient law?
The log of a quotient is the difference of the logs: ln(a/b) = ln a − ln b (for a, b > 0). For example, ln(10/2) = ln 10 − ln 2 = ln 5.
Why isn’t ln(a/b) equal to (ln a)/(ln b)?
Test a = b = e: ln(e/e) = ln 1 = 0, but (ln e)/(ln e) = 1/1 = 1. Division inside becomes subtraction outside, not division.
Can I split ln(a − b) into ln a − ln b?
Never. ln(5−3) = ln 2 ≈ 0.69, but ln 5 − ln 3 ≈ 0.51. The law splits quotients, not differences.
Why does division become subtraction?
Because a/b = a·b−1: the product law turns it into ln a + ln(b−1), and ln(b−1) = −ln b. It is also the mirror of ex/ey = ex−y.
What is ln(1/x)?
−ln x. By the law: ln(1/x) = ln 1 − ln x = 0 − ln x = −ln x. This is the form you’ll use most — e.g. d/dx [ln(1/x)] = −1/x.
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