Step 1 of 10
Read the seal
Read the fraction structure before opening the logarithm.
Worked example · real topic question
log ((a√b) / c2)
log ((x3y4) / z6)
logb ((√x · ∛(y2)) / ⁵√(z2))
log ((a2b3) / c4)
Read structure
Split operations
Lift powers
Check signs
Logarithm Expansion Deck
One sealed logarithm. Two structural lanes.
Assume the logarithm is defined. First identify what is above and below the fraction bar.
log ((a√b) / c2)do not move exponents yet
Numerator: a · √b
will become positive log terms
will become positive log terms
Denominator: c2
will become a subtracted log term
will become a subtracted log term
QUOTIENT LAW FIRST · log(M/N) = log M − log N
Numerator lane · keep positive
log(a√b)
−denominator
crosses here
crosses here
Denominator lane · subtract
log(c2)
PRODUCT LAW NEXT · log(MN) = log M + log N
log a
+
log(√b)
−
log(c2)
POWER LAW LAST · log(Mk) = k log M
log a
+
log b
−
log c
log a + 12log b − 2log c
The denominator creates the minus sign. The exponent creates the coefficient.
Run the same deck with three powers.
Split the quotient, split the numerator product, then lift every exponent.
SPLIT OPERATIONS · numerator terms stay + · denominator term becomes −
Numerator lane
log(x3) + log(y4)
−quotient
sign
sign
Denominator lane
log(z6)
LIFT POWERS · each exponent becomes its log coefficient
log x
+
log y
−
log z
3log x + 4log y − 6log z
Do not multiply 3 and 4. Each power belongs to its own logarithm.
Roots are powers wearing radical symbols.
Rewrite mentally: √x = x1/2, ∛(y2) = y2/3, and ⁵√(z2) = z2/5.
ROOT INDEX → EXPONENT DENOMINATOR · keep the quotient sign
1/2logb x
+ 2/3logb y
− 2/5logb z
12logb x + 23logb y − 25logb z
The fifth root produces 2/5. Its denominator position produces the minus.
Your turn: issue the coefficient receipt.
Enter the signed coefficient of each logarithm. Include the negative sign for the denominator term.
log ((a2b3) / c4)
Read the fraction structure, then lift the three powers.
Use your whiteboard.Expand the logarithm fully. Keep the numerator terms positive and the denominator term negative. Submit when your final line is ready.
Work submitted — the signed receipt is unlocked.
Check the direction in reverse.
Coefficients return to powers; plus joins a product; minus puts a factor in the denominator.
EXPAND → CONDENSE CHECK · every sign must return to its original lane
log a
+
log b
−
log c
2log a + 3log b − 4log c
↔
log ((a2b3) / c4)
The reverse check recreates the original expression exactly, so the coefficients and signs are correct.
log
Expansion deck cleared.
One structure-reading routine handles products, quotients, powers, and roots.
1Read numerator and denominator
2Split quotient, then products
3Lift powers into coefficients
4Reverse-check every sign