Algebra 2 Worksheets | Exponential and Logarithmic Functions ... - Free Printable

Let’s solve each problem step by step. We’ll use the basic properties of logarithms:

Key Logarithm Rules (remember these!):

1. Product Rule:
logₐ(x·y) = logₐx + logₐy

2. Quotient Rule:
logₐ(x/y) = logₐx – logₐy

3. Power Rule:
logₐ(xⁿ) = n·logₐx

4. Condensing: Reverse of above — combine logs into one using the same rules.

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Expand each logarithm:

1) log₄(7 · 9 · 3)
Use Product Rule: break apart multiplication inside log.
= log₄7 + log₄9 + log₄3

✔ Final Answer for #1: log₄7 + log₄9 + log₄3

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2) log₁₆(p · s · d)^(1/4)
First, Power Rule: bring exponent out front.
= (1/4) · log₁₆(p · s · d)
Then Product Rule:
= (1/4)(log₁₆p + log₁₆s + log₁₆d)

✔ Final Answer for #2: (1/4)(log₁₆p + log₁₆s + log₁₆d)

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3) log₆(m/s)
Quotient Rule:
= log₆m – log₆s

✔ Final Answer for #3: log₆m – log₆s

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4) log₉(s⁶/m)
Break it up: Quotient first, then Power on numerator.
= log₉(s⁶) – log₉m
= 6·log₉s – log₉m

✔ Final Answer for #4: 6log₉s – log₉m

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5) log₁₂(d · m)²
Power Rule first:
= 2 · log₁₂(d · m)
Then Product Rule:
= 2(log₁₂d + log₁₂m)

✔ Final Answer for #5: 2(log₁₂d + log₁₂m)

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6) log₅(s⁵ · b)
Product Rule first:
= log₅(s⁵) + log₅b
Then Power Rule:
= 5log₅s + log₅b

✔ Final Answer for #6: 5log₅s + log₅b

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Condense each expression to one logarithm:

7) 3log₁₁z – 3log₁₁w
Factor out the 3? Or apply Power Rule first:
= log₁₁(z³) – log₁₁(w³) → now Quotient Rule
= log₁₁(z³ / w³)

Alternatively: factor 3 first → 3(log₁₁z – log₁₁w) = 3log₁₁(z/w) = log₁₁((z/w)³) — same thing!

✔ Final Answer for #7: log₁₁(z³/w³) or log₁₁((z/w)³) — both are correct. Let’s write as log₁₁(z³/w³)

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8) 6log₈9 + 3log₈7
Apply Power Rule:
= log₈(9⁶) + log₈(7³)
Now Product Rule:
= log₈(9⁶ · 7³)

✔ Final Answer for #8: log₈(9 · 7³)

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9) log4 + log3
Assume base 10 if not written.
Product Rule:
= log(4·3) = log12

✔ Final Answer for #9: log12

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10) log₆7 – 2log₆9
Power Rule on second term:
= log₆7 – log₆(9²) = log₆7 – log₆81
Quotient Rule:
= log₆(7/81)

✔ Final Answer for #10: log₆(7/81)

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11) 6log₃4 + 4log₃6 + log₃3
Apply Power Rule to each:
= log₃(4⁶) + log₃(6⁴) + log₃(3¹)
Now add them all with Product Rule:
= log₃(4⁶ · 6⁴ · 3)

We can leave it like this — no need to compute actual numbers.

✔ Final Answer for #11: 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 Answer for #12: log₁₅(6² · 9⁶ · 2⁴)

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## ✔ FINAL ANSWERS:

1) log₄7 + log₄9 + log₄3
2) (1/4)(log₁₆p + log₁₆s + log₁₆d)
3) log₆m – log₆s
4) 6log₉s – log₉m
5) 2(log₁₂d + log₁₂m)
6) 5log₅s + log₅b
7) log₁₁(z³/w³)
8) log₈(9⁶ · 7³)
9) log12
10) log₆(7/81)
11) log₃(4⁶ · 6⁴ · 3)
12) log₁₅(6² · 9⁶ · 2⁴)