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2.12 · Product Property
State the Product Property of Logarithms.
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2.12 · Quotient Property
State the Quotient Property of Logarithms.
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2.12 · Power Property
State the Power Property of Logarithms.
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2.12 · Change of Base
State the Change of Base formula. What does it tell us about all log functions?
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2.12 · Condense
Condense to a single log: log₂ x + log₂ y
Which property applies?
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2.12 · Condense
Condense to a single log: 2 log₃ x − log₃ y
Apply Power first, then Quotient
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2.12 · Condense
Condense: 2 log₇ a − 5 log₇ b + log₇ 4
Three steps: Power, then Product, then Quotient
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2.12 · Expand
Expand: log(36x) using the Product Property.
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✓ Quotient Property
logb(x/y) = logb x − logb y
Dividing inside a log → subtracting logs. Example: log₃(x/4) = log₃ x − log₃ 4.
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✓ Product Property
logb(xy) = logb x + logb y
Multiplying inside a log → adding logs. Example: log₂(5x) = log₂ 5 + log₂ x.
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✓ Change of Base
logb x = loga x / loga b
All log functions are vertical dilations of each other. Commonly: log₄ x = log x / log 4 = ln x / ln 4.
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✓ Power Property
logb(xn) = n · logb x
Exponent inside ↔ coefficient outside. Example: log₇(x³) = 3·log₇ x. Also: 3·log x = log(x³).
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✓ Condense: 2 log₃ x − log₃ y
log₃(x²/y)
Power: 2 log₃ x = log₃(x²). Quotient: log₃(x²) − log₃ y = log₃(x²/y).
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✓ Condense: log₂ x + log₂ y
log₂(xy)
Product Property: adding logs → multiply inside. log₂ x + log₂ y = log₂(xy).
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✓ Expand: log(36x)
log 36 + log x
Product Property in reverse: log(36x) = log 36 + log x. Note: log 36 = log(6²) = 2 log 6 ≈ 1.556.
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✓ Condense: 2a − 5b + 4
log₇(4a²/b&sup5;)
Power: log₇(a²), log₇(b&sup5;). Product with 4: log₇(4a²). Quotient: log₇(4a²/b&sup5;).
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2.12 · Natural Log
Condense: 3 ln x − 4 ln y
Same properties apply to ln
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2.12 · Vertical Translation
f(x) = log(6x) and g(x) = log x. Write f(x) as a vertical translation of g(x).
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2.12 · Common Mistakes
True or False: log(x + y) = log x + log y
Think carefully about the Product Property
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2.12 · Condense
Condense: log₁₀ x − log₁₀ 5 − log₁₀ z
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✓ Vertical Translation
f(x) = g(x) + log 6
log(6x) = log 6 + log x = g(x) + log 6. k = log 6 ≈ 0.778. A horizontal dilation ≡ a vertical translation!
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✓ Condense: 3 ln x − 4 ln y
ln(x³/y⁴)
Power: 3 ln x = ln(x³) and 4 ln y = ln(y⁴). Quotient: ln(x³) − ln(y⁴) = ln(x³/y⁴).
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✓ Condense
log(x / 5z)
Quotient: log x − log 5 = log(x/5). Then − log z: log(x/5) − log z = log(x/5z).
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✓ Common Mistake: FALSE
FALSE — log(x + y) ≠ log x + log y
The Product Property says log(xy) = log x + log y (multiplication inside, not addition). Logs do NOT distribute over addition.