Let’s work through each problem step by step.
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Part 1: Expand each logarithm
We use these rules:
- log(a·b) = log a + log b
- log(a/b) = log a - log b
- log(a^n) = n·log a
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1) log₄(7·9·3)
Break the product into separate logs:
= log₄7 + log₄9 + log₄3
✔ Final:
log₄7 + log₄9 + log₄3
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2) log₁₆(p·s·d)^(1/4)
First, bring the exponent out front:
= (1/4)·log₁₆(p·s·d)
Now expand the product inside:
= (1/4)(log₁₆p + log₁₆s + log₁₆d)
✔ Final:
(1/4)(log₁₆p + log₁₆s + log₁₆d)
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3) log₆(m/s)
Use quotient rule:
= log₆m - log₆s
✔ Final:
log₆m - log₆s
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4) log₅(s⁶/m)
Quotient rule first:
= log₅(s⁶) - log₅m
Then power rule on s⁶:
= 6·log₅s - log₅m
✔ Final:
6·log₅s - log₅m
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5) log₁₂(d·m)²
Power rule first:
= 2·log₁₂(d·m)
Now expand the product:
= 2(log₁₂d + log₁₂m)
✔ Final:
2(log₁₂d + log₁₂m)
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6) log₉(s⁵·b)
Product rule:
= log₉(s⁵) + log₉b
Power rule on s⁵:
= 5·log₉s + log₉b
✔ Final:
5·log₉s + log₉b
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Part 2: Condense each expression to one logarithm
Reverse the rules:
- log a + log b = log(a·b)
- log a - log b = log(a/b)
- n·log a = log(a^n)
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7) 3log₁₁z - 3log₁₁w
Factor out the 3:
= 3(log₁₁z - log₁₁w)
Quotient rule inside:
= 3·log₁₁(z/w)
Power rule:
= log₁₁((z/w)³)
✔ Final:
log₁₁((z/w)³) or
log₁₁(z³/w³)
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8) 6log₈9 + 3log₈7
Power rule first:
= log₈(9⁶) + log₈(7³)
Product rule:
= log₈(9⁶ · 7³)
✔ Final:
log₈(9⁶ · 7³)
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9) log4 + log3
Same base? Yes (base 10 assumed if not written).
Product rule:
= log(4·3) = log(12)
✔ Final:
log12
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10) log₈7 - 2log₈9
Power rule on second term:
= log₈7 - log₈(9²) = log₈7 - log₈81
Quotient rule:
= log₈(7/81)
✔ Final:
log₈(7/81)
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11) 6log₃4 + 4log₃6 + log₃3
Power rule on each:
= log₃(4⁶) + log₃(6⁴) + log₃(3)
Product rule all together:
= log₃(4⁶ · 6⁴ · 3)
✔ Final:
log₃(4⁶ · 6⁴ · 3)
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12) 2log₁₅6 + 6log₁₅9 + 4log₁₅2
Power rule:
= log₁₅(6²) + log₁₅(9⁶) + log₁₅(2⁴)
Product rule:
= log₁₅(6² · 9⁶ · 2⁴)
✔ Final:
log₁₅(6² · 9⁶ · 2⁴)
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Final Answer:
1) log₄7 + log₄9 + log₄3
2) (1/4)(log₁₆p + log₁₆s + log₁₆d)
3) log₆m - log₆s
4) 6·log₅s - log₅m
5) 2(log₁₂d + log₁₂m)
6) 5·log₉s + log₉b
7) log₁₁((z/w)³)
8) log₈(9⁶ · 7³)
9) log12
10) log₈(7/81)
11) log₃(4⁶ · 6⁴ · 3)
12) log₁₅(6² · 9⁶ · 2⁴)
Parent Tip: Review the logic above to help your child master the concept of properties of logs worksheet.
