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Amy Ferguson Moncure

H Algebra 2/Trig · Unit 7

Expand Logarithmic Expressions

Rewrite one logarithm as a sum or difference. A root becomes a fraction coefficient. A plain number log that you know exactly should be written as that number.

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Key Ideas Easy Medium Hard

log(MN) = log M + log N

log of a quotient = log M − log N

log(M to the c) = c log M

What to Remember

One log becomes a sum or a difference

A product inside the log becomes a sum. A quotient becomes a difference.

An exponent inside becomes a coefficient.

Roots become coefficients

square root of x is x to the power 12, so it contributes a coefficient of 12.

A cube root of x2 contributes a coefficient of 23.

A plain number may simplify

log5 25 = 2. When you expand, write that 2, not log5 25.

Level 1

Easy

Products, quotients, and whole-number powers

Example
Expand log2 (x³y5).

Quotient rule: a quotient becomes a difference.

log2 x³y − log2 5

Product rule, then the power rule.

3 log2 x + log2 y − log2 5

Answer: 3 log2 x + log2 y − log2 5

1
Expand log3 ab.
Show Answer

Product rule.

log3 a + log3 b

Answer: log3 a + log3 b

2
Expand log5 (mn).
Show Answer

Quotient rule.

log5 m − log5 n

Answer: log5 m − log5 n

3
Expand log7 p⁴.
Show Answer

Power rule: an exponent becomes a coefficient.

4 log7 p

Answer: 4 log7 p

4
Expand ln(4t²).
Show Answer

Product rule.

ln 4 + ln(t²)

Power rule.

ln 4 + 2 ln t

Answer: ln 4 + 2 ln t

5
Expand log xyz.
Show Answer

log with no base means log10.

log x + log y − log z

Answer: log x + log y − log z

Level 2

Medium

A square root or a cube root

Example
Expand log 6a²square root of bc.

Quotient rule.

log 6a²square root of b − log c

Product rule and power rule.

log 6 + 2 log a + 12 log b − log c

Answer: log 6 + 2 log a + 12 log b − log c

1
Expand log4 (x³square root of y).
Show Answer

Quotient rule, then power rule.

3 log4 x − 12 log4 y

Answer: 3 log4 x − 12 log4 y

2
Expand log2 (square root of ab³).
Show Answer

12 log2 a − 3 log2 b

Answer: 12 log2 a − 3 log2 b

3
Expand ln(p²\(\sqrt[3]{q}\)).
Show Answer

Product rule.

2 ln p + ln(\(\sqrt[3]{q}\))

A cube root is a power of 13.

2 ln p + 13 ln q

Answer: 2 ln p + 13 ln q

4
Expand log5 (25x⁴y).
Show Answer

log5 25 + 4 log5 x − log5 y

log5 25 = 2

2 + 4 log5 x − log5 y

Answer: 2 + 4 log5 x − log5 y

5
Expand log3 (a²bc⁵).
Show Answer

2 log3 a + log3 b − 5 log3 c

Answer: 2 log3 a + log3 b − 5 log3 c

Level 3

Hard

A rational power and a constant that simplifies

Example
Expand log2 (7\(\sqrt[3]{m^{2}}\)n⁴p).

Quotient rule.

log2 7\(\sqrt[3]{m^{2}}\) − log2 n⁴p

The cube root of m² is m to the power 23.

log2 7 + 23 log2 m − 4 log2 n − log2 p

Answer: log2 7 + 23 log2 m − 4 log2 n − log2 p

1
Expand log5 (3\(\sqrt[3]{x^{2}}\)yz⁴).
Show Answer

log5 3 + 23 log5 x − log5 y − 4 log5 z

Answer: log5 3 + 23 log5 x − log5 y − 4 log5 z

2
Expand log4 (square root of x · y³8w²).
Show Answer

12 log4 x + 3 log4 y − log4 8 − 2 log4 w

8 = 2³ and 4 = 2², so log4 8 = 32.

12 log4 x + 3 log4 y − 32 − 2 log4 w

Answer: 12 log4 x + 3 log4 y − 32 − 2 log4 w

3
Expand ln e²square root of xy³.
Show Answer

ln(e²) + ln(square root of x) − ln(y³)

ln(e²) = 2

2 + 12 ln x − 3 ln y

Answer: 2 + 12 ln x − 3 ln y

4
Expand log6 (a⁵square root of b · c²).
Show Answer

5 log6 a − 12 log6 b − 2 log6 c

Answer: 5 log6 a − 12 log6 b − 2 log6 c

5
Expand log3 (9\(x^{\frac{2}{3}}\)yw⁴).
Show Answer

log3 9 + 23 log3 x − log3 y − 4 log3 w

log3 9 = 2

2 + 23 log3 x − log3 y − 4 log3 w

Answer: 2 + 23 log3 x − log3 y − 4 log3 w

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