A logarithm answers: "to what power must we raise a base to get this result?" It is the inverse of exponentiation — and the tool that turns multiplication into addition.
If by = x, then logb(x) = y.
| Symbol | Meaning | Constraint |
|---|---|---|
b |
Base | b > 0 and b ≠ 1 |
x |
Argument | x > 0 — you cannot log zero or negatives |
y |
Exponent (the answer) | Can be negative, zero, or fractional |
Example: Since 23 = 8, then log2(8) = 3. Read
it as "2 to what power gives 8?"
Inverse intuition:
blogb(x) = x and
logb(by) = y — they undo each other. This is why logs turn
hard multiplicative problems (pH, Richter, compound interest) into additive ones.
These come directly from exponent laws —
bm × bn = bm+n becomes addition in log-space.
logb(M × N) = logb(M) + logb(N)
Example:
log2(32) = log2(4 × 8) = log2(4) + log2(8) = 2
+ 3 = 5.
logb(M / N) = logb(M) − logb(N)
Example:
log10(1000) − log10(10) = log10(100) = 2.
logb(Mp) = p × logb(M)
Example:
log2(82) = 2 × log2(8) = 2 × 3 = 6
(indeed 82=64=26).
logb(b) = 1 — since
b1 = b
logb(1) = 0 — since
b0 = 1
blogb(x) = x — exponentiation undoes the log
log(M + N) ≠ log M + log N and log(M − N) ≠ log M − log N. Only ×/÷
inside becomes +/− outside. Also: log(Mp) brings the
p down, but (log M)p does not — parentheses matter.
| Type | Notation | Base | Used for |
|---|---|---|---|
| Common | log(x) or log10(x) |
10 | Richter scale, pH, decibels — base-10 is human-friendly |
| Natural | ln(x) |
e ≈ 2.718 |
Calculus, continuous growth/decay — calculus simplifies with
e
|
logb(expression) = y → convert to exponential:
expression = by, then solve.
log3(x + 1) = 2x + 1 = 32 = 9 → x = 8x+1 = 9 > 0 ✓ valid.
logb(A) = logb(B) → same base ⇒ arguments equal:
A = B (provided both > 0).
log5(2x) = log5(10)2x = 10 → x = 52x = 10 > 0 ✓ valid.
Use product/quotient rules to combine logs into a single logarithm on each side, then use Type A or B.
log2(x) + log2(x − 2) = 3log2[x(x − 2)] = 3 (product rule)x(x − 2) = 23 = 8x2 − 2x − 8 = 0 → (x − 4)(x + 2) = 0 →
x = 4 or x = −2log2(x) needs x > 0 →
reject −2. Also x − 2 > 0 needs x > 2.x = 4 only.
Calculators only have log10 and ln. To compute any base:
logb(x) = logk(x) / logk(b)
— usually k = 10 or e.
Example:
log5(20) = log10(20) / log10(5) ≈ 1.301 / 0.699 ≈
1.861.
When to use: any time the base is not 10 or e but you have a calculator. Also
useful for comparing logs: log2(10) vs log3(10) — convert
both to base 10 and compare.
log10(0.01) = log10(10−2) = −2
(power rule). Any number between 0 and 1 has a negative log; numbers > 1 have positive logs
(if base > 1). This is why pH (which logs small H⁺ concentrations) is positive via a minus
sign:
pH = −log[H⁺].
You must test every candidate solution in the original equation's arguments. Squaring or combining logs can introduce values that make an argument ≤ 0.
x = −5 but original has log(x) or
log(x−2), that argument is negative → reject.
log x + log(x−2) implicitly assumes x > 2. The
algebra may return x = −2 — mathematically a root of the quadratic but not in
the domain.
| Symbol | Meaning |
|---|---|
b |
Base ( > 0, ≠ 1 ) |
x, M, N |
Arguments (must be > 0) |
y, p |
Exponents / powers |
If 2³ = 8, then log₂(8) = ?
log_b(M × N) = ?
log₂(8²) = ?
log_b(b) and log_b(1) are respectively?
log₃(x+1)=2 → x = ?
log₂(x)+log₂(x−2)=3 → valid x?
Change of base: log₅(20) = ?
log₁₀(0.01) = ?
by=x ⇒ logb(x)=y ; b >0, b≠1 ; x >0
Inverse ⇒ blogb(x)=x ; logb(by)=y
Product ⇒ log(M×N)=logM+logN
Quotient ⇒ log(M/N)=logM−logN
Power ⇒ log(Mp)=p·logM (bring power down)
Identity ⇒ logb(b)=1 ; logb(1)=0 ; blogb(x)=x
Common trap ⇒ log(M+N)≠logM+logN ×
Common ⇒ log₁₀ ; Natural ⇒ ln (base e≈2.718)
a) logb(expr)=y ⇒ expr=by (eg log₃(x+1)=2→x=8)
b) logb(A)=logb(B) ⇒ A=B (same base)
c) Multiple logs ⇒ combine via product/quotient → Type A/B
(eg log₂x+log₂(x−2)=3→x(x−2)=8→x=4, −2× rejected)
Change of base ⇒ logb(x)=logk(x)/logk(b)
Decimal ⇒ log₁₀(0.01)=log₁₀(10⁻²)=−2 ; 0<x<1 → negative log
Extraneous ⇒ must check original args >0 ; reject negatives/zero
b ⇒ base ; x,M,N ⇒ arguments ; y,p ⇒ exponents
Primary source: none pinned yet — drop your preferred video/book resource into RESOURCES.md and it will be linked here. Ask me anything that's unclear.
Questions? Ask your agent — you can follow up on any concept, quiz answer, or get extra practice problems tuned to this module.