Condense Logarithmic Expressions

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Key Ideas Easy Medium Hard

log M + log N = log(MN)

log M − log N = log(of a quotient)

c log M = log of M to the c

What to Remember

One log at the end

Add logs: multiply the arguments. Subtract logs: divide the arguments.

A coefficient becomes an exponent.

Roots

A coefficient of 12 is a square root. A coefficient of 13 is a cube root.

Use square root of x for a square root.

Constants can simplify

If a log of a number is itself a whole number, multiply or divide that number before you stop.

12 log₃ 9 = log₃ 3 = 1.

Level 1

Easy

Two or three logs

Example

Condense 2 log₃ x + log₃ y − log₃ 5.

Coefficient rule: c log_b M = log_b(M to the c).

2 log₃ x = log₃ x²

Product rule: add logs with the same base by multiplying the arguments.

log₃ x² + log₃ y = log₃ x²y

Quotient rule: subtract logs with the same base by dividing the arguments.

log₃ x²y − log₃ 5 = log₃ (x²y5)

Answer: log₃ (x²y5)

Condense log₂ a + log₂ b.

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Product rule: add logs with the same base by multiplying the arguments.

log₂ a + log₂ b

log₂ ab

Answer: log₂ ab

Condense log₅ m − log₅ n.

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Quotient rule: subtract logs with the same base by dividing the arguments.

log₅ m − log₅ n

log₅ (mn)

Answer: log₅ (mn)

Condense 3 log₇ p.

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Coefficient rule: c log_b M = log_b(M to the c).

3 log₇ p

log₇ p³

Answer: log₇ p³

Condense log x + log y − log z.

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log with no base means log₁₀.

Product rule, then quotient rule.

log xy − log z

log xyz

Answer: log xyz

Condense 2 ln t + ln 4.

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Coefficient rule.

2 ln t = ln(t²)

Product rule.

ln(t²) + ln 4 = ln(4t²)

Answer: ln(4t²)

Level 2

Medium

A fraction coefficient or a root

Example

Condense 2 log₅ x + 12 log₅ y − log₅ 3.

Coefficient rule.

2 log₅ x = log₅ x²

12 log₅ y = log₅ (square root of y)

Multiply, then divide.

log₅ (x²square root of y3)

Answer: log₅ (x²square root of y3)

Condense 3 log₂ a − 2 log₂ b + log₂ c.

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3 log₂ a = log₂ a³

2 log₂ b = log₂ b²

log₂ (a³cb²)

Answer: log₂ (a³cb²)

Condense ln 5 + 13 ln x − 2 ln y.

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13 ln x = ln(\(x^{\frac{1}{3}}\))

2 ln y = ln(y²)

ln 5\(x^{\frac{1}{3}}\)y²

Answer: ln 5 \(x^{\frac{1}{3}}\)y²

Condense 4 log₃ x + 12 log₃ 9 − log₃ y.

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12 log₃ 9 = log₃ (square root of 9)

square root of 9 = 3

12 log₃ 9 = log₃ 3

4 log₃ x = log₃ x⁴

log₃ (3x⁴y)

Answer: log₃ (3x⁴y)

Condense log₇ 4 + log₇ x − 2 log₇ y.

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2 log₇ y = log₇ y²

log₇ (4xy²)

Answer: log₇ (4xy²)

Condense 2 log₂ 8 + log₂ x − log₂ 4.

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2 log₂ 8 = log₂ 8²

8² = 64

log₂ 64 − log₂ 4 = log₂ (644)

644 = 16

log₂ 16x

Answer: log₂ 16x

Level 3

Hard

Four logs, and constants that simplify

Example

Condense ln 9 − 2 ln p + 12 ln q − ln 3.

2 ln p = ln(p²)

12 ln q = ln(square root of q)

ln 9square root of q3p²

93 = 3

ln 3square root of qp²

Answer: ln 3square root of qp²

Condense 2 log₅ x + log₅ 10 − 3 log₅ y − log₅ 2.

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2 log₅ x = log₅ x²

3 log₅ y = log₅ y³

log₅ (10x²2y³)

102 = 5

log₅ (5x²y³)

Answer: log₅ (5x²y³)

Condense 3 log₄ a + 12 log₄ b − log₄ c − 2 log₄ a.

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3 log₄ a − 2 log₄ a = log₄ a

12 log₄ b = log₄ (square root of b)

log₄ (asquare root of bc)

Answer: log₄ (asquare root of bc)

Condense ln 8 + 13 ln x − ln 2 − 2 ln y.

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ln 8 − ln 2 = ln 82

82 = 4

13 ln x = ln(\(x^{\frac{1}{3}}\))

2 ln y = ln(y²)

ln 4\(x^{\frac{1}{3}}\)y²

Answer: ln 4 \(x^{\frac{1}{3}}\)y²

Condense log₆ 18 − log₆ 3 + 2 log₆ x − 12 log₆ y.

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log₆ 18 − log₆ 3 = log₆ (183)

183 = 6

2 log₆ x = log₆ x²

12 log₆ y = log₆ (square root of y)

log₆ (6x²square root of y)

Answer: log₆ (6x²square root of y)

Condense 4 log₃ m + 12 log₃ 9 − log₃ n − log₃ 3.

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12 log₃ 9 = log₃ 3

log₃ 3 − log₃ 3 = 0

4 log₃ m = log₃ m⁴

log₃ (m⁴n)

Answer: log₃ (m⁴n)

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