A log is an exponent, so the log laws are the exponent laws wearing different clothes.
Expand
Expand log x²yz
A product becomes a sum, a quotient becomes a difference, and a power drops to the front: log x² = 2 log x. So it expands to 2 log x + log y − log z. C treats log like a multiplier — but log(ab) is log a + log b, never log a × log b.
| Inside the log | Outside | Because exponents |
|---|---|---|
| logb(MN) | log M + log N | multiply → add |
| logb(MN) | log M − log N | divide → subtract |
| logb(Mk) | k · log M | power → coefficient |
Everything drops one level: multiplying becomes adding, powers become multipliers.
Two more worth knowing: logb1 = 0, since b⁰ = 1, and logbb = 1, since b¹ = b.
log (x²y ÷ z)
the division is outermost
log(x²y) − log z
the quotient law
log x² + log y − log z
the product law
2 log x + log y − log z
the power law
If the bottom were yz, the answer would be −log y − log z, not −log y + log z. Put brackets in first if that sign is at all unclear.
2 log x + 3 log y − log z
the expression
log x² + log y³ − log z
coefficients become powers
log(x²y³) − log z
plus becomes times
log x²y³z
minus becomes divide
Deal with the coefficients first. Trying to combine while a 2 or 3 is still outside never works.
In 2 log₃ x − 2 + log₃ y, write the 2 as log₃ 9 (since 3² = 9). Then it condenses: log₃ x² − log₃ 9 + log₃ y = log₃ (x²y9). Never tack the number on inside the log as “+ 2”.
Condensing also solves an equation. log₂ x + log₂ (x + 2) = 3 becomes log₂ (x(x + 2)) = 3, so x(x + 2) = 2³ = 8, x² + 2x − 8 = 0, (x + 4)(x − 2) = 0. Reject x = −4 (the log of a negative number does not exist), so x = 2.
Your calculator only has log base 10 and ln. That is enough.
logb a = log alog b
log₃ 20 = log 20 ÷ log 3
any base on top and bottom
= 1.3010 ÷ 0.4771
from the calculator
≈ 2.727
3² = 9, 3³ = 27 — fits
For exact values, skip it. log₂32 asks “2 to what power is 32?” — the answer is 5, no calculator required.
A fraction inside gives a negative answer: log₂ (18) asks “2 to what power is 18?”, and 2⁻³ = 18, so it is −3.
Let log 2 = a and log 3 = b. Express log 18, log (83) and log 5 in a and b.
18 = 2 × 3² → log 18 = log 2 + 2 log 3 = a + 2b
factor into primes first
83 = 2³ ÷ 3 → log (83) = 3 log 2 − log 3 = 3a − b
divide → subtract
5 = 10 ÷ 2 → log 5 = log 10 − log 2 = 1 − a
a 5 hides in 10, and log 10 = 1
log 18 is a + 2b, never a · b² or 2ab. The letters stand for logs, so they follow the log laws: they add and get coefficients; they never multiply.
Evaluate log₁₆ 8, log₂₅ 125, and log₂ 5 × log₅ 16.
log₁₆ 8 = k means 16k = 8
turn it into an exponent question
24k = 2³ → k = 34
common base 2
25k = 125 → 52k = 5³ → k = 32
common base 5
(log 5 ÷ log 2)(log 16 ÷ log 5) = log 16 ÷ log 2
change of base; the log 5 cancels
= log₂ 16 = 4
the chain collapses
Shortcut: logbm (bn) = n/m. So log₁₆ 8 = log2⁴ 2³ = 34. Check by powering back: 163/4 = (⁴√16)³ = 2³ = 8 ✓.
A root, a number and a product below the bar. Expand log₂ (√x ÷ 8y).
log₂ x1/2 − log₂ (8y)
√x is x to the ½; the whole bottom is subtracted
½ log₂ x − (log₂ 8 + log₂ y)
split the product inside a bracket
½ log₂ x − 3 − log₂ y
log₂ 8 = 3 because 2³ = 8
What is different: a root becomes a fraction coefficient, and a plain number inside the log comes out as a number.
A decimal. With log 2 = a and log 3 = b, express log 0.36.
0.36 = 36 ÷ 100 = 2² × 3² ÷ 10²
a decimal is a fraction over a power of 10
log 0.36 = 2 log 2 + 2 log 3 − 2 log 10
powers to coefficients
= 2a + 2b − 2
log 10 = 1
What is different: the answer is negative (about −0.44), as every log of a number between 0 and 1 is.
A base below 1. Evaluate log13 27.
(13)k = 27
“what power?”
3−k = 3³
13 = 3−1
k = −3
check: (13)−3 = 3³ = 27 ✓
What is different: with a base below 1, a big number has a negative log. Writing 3 drops the minus that 13 carries.
| Tempting | Truth |
|---|---|
| log(M + N) = log M + log N | false |
| log M ÷ log N = log(M − N) | false |
| (log M)² = 2 log M | false |
| log(MN) = log M + log N | true |
| log(Mk) = k log M | true |
log(x²) is 2 log x. But (log x)² is log x multiplied by itself, and no law touches it. Where the bracket sits changes everything.
Test any suspicious “law” with numbers: log(10 + 10) = log 20 ≈ 1.3, while log 10 + log 10 = 2. Not equal.
The usual shapes: condense to one log (watch the lone number), express a log in terms of log 2 and log 3, or evaluate an exact log across bases. Turn every exact log into “what power?”, then check the answer by raising the base to it.
Let x = 10, y = 10, z = 10, base 10.
Condense
Condense ½ log x + 2 log y − 3 log z
Each coefficient climbs back up as a power — ½ becomes a square root, 2 a square, 3 a cube — then + becomes × and − becomes ÷: log(√x · y² ÷ z³). C multiplies by the coefficients instead of raising to them.
