Solved Solving Logarithm Equations Worksheet Solve each | Chegg.com - Free Printable

Let’s solve each equation one by one. We’ll use the definition of logarithms and basic algebra.

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1) 9 log₉ v = 0

Divide both sides by 9:
log₉ v = 0

Convert to exponential form:
v = 9⁰ = 1

✔ Answer: v = 1

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2) –log₉ n = 1

Multiply both sides by –1:
log₉ n = –1

Exponential form:
n = 9⁻¹ = 1/9

✔ Answer: n = 1/9

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3) –7 – 10 log₆ r = –27

Add 7 to both sides:
–10 log₆ r = –20

Divide by –10:
log₆ r = 2

Exponential form:
r = 6² = 36

✔ Answer: r = 36

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4) 7 log₅ x – 4 = 17

Add 4 to both sides:
7 log₅ x = 21

Divide by 7:
log₅ x = 3

Exponential form:
x = 5³ = 125

✔ Answer: x = 125

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5) –4 log₆ (–r) = –4

Divide both sides by –4:
log₆ (–r) = 1

Exponential form:
–r = 6¹ → r = –6

Check domain: log₆(–r) = log₆(6) → valid since –r = 6 > 0

✔ Answer: r = –6

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6) –4 + log₂ (–8p) = –3

Add 4 to both sides:
log₂ (–8p) = 1

Exponential form:
–8p = 2¹ = 2 → p = –2/8 = –1/4

Check domain: –8p = –8*(–1/4) = 2 > 0 → valid

✔ Answer: p = –1/4

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7) 4 – 8 log₇ (2x) = –28

Subtract 4:
–8 log₇ (2x) = –32

Divide by –8:
log₇ (2x) = 4

Exponential form:
2x = 7⁴ = 2401 → x = 2401 / 2 = 1200.5

✔ Answer: x = 1200.5 or 2401/2

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8) 6 + 3 log₅ (k – 6) = 15

Subtract 6:
3 log₅ (k – 6) = 9

Divide by 3:
log₅ (k – 6) = 3

Exponential form:
k – 6 = 5³ = 125 → k = 131

✔ Answer: k = 131

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9) 9 log₃ (–5r – 3) = 36

Divide by 9:
log₃ (–5r – 3) = 4

Exponential form:
–5r – 3 = 3⁴ = 81

Solve:
–5r = 84 → r = –84/5 = –16.8

Check domain: –5r – 3 = –5*(-16.8) – 3 = 84 – 3 = 81 > 0 → valid

✔ Answer: r = –84/5 or –16.8

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10) log₆ (9 – 7x) – 7 = –6

Add 7:
log₆ (9 – 7x) = 1

Exponential form:
9 – 7x = 6¹ = 6 → –7x = –3 → x = 3/7

Check domain: 9 – 7*(3/7) = 9 – 3 = 6 > 0 → valid

✔ Answer: x = 3/7

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11) 9 log₆ (2a + 1) + 6 = 33

Subtract 6:
9 log₆ (2a + 1) = 27

Divide by 9:
log₆ (2a + 1) = 3

Exponential form:
2a + 1 = 6³ = 216 → 2a = 215 → a = 215/2 = 107.5

✔ Answer: a = 215/2 or 107.5

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12) –3 + 8 log₉ (3x + 7) = 29

Add 3:
8 log₉ (3x + 7) = 32

Divide by 8:
log₉ (3x + 7) = 4

Exponential form:
3x + 7 = 9⁴ = 6561 → 3x = 6554 → x = 6554 / 3 ≈ 2184.666...

But let’s keep it exact:
x = 6554 / 3

Check domain: 3x + 7 = 6561 > 0 → valid

✔ Answer: x = 6554/3

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13) log₁₅ (4 – p) = log₁₅ (–2p + 2)

Since logs are equal and same base, arguments must be equal:
4 – p = –2p + 2

Solve:
4 – p = –2p + 2 → add 2p to both sides:
4 + p = 2 → p = –2

Check domain:
Left: 4 – (–2) = 6 > 0
Right: –2*(–2) + 2 = 4 + 2 = 6 > 0 → valid

✔ Answer: p = –2

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14) log₂ (–4x + 2) = log₂ (5x + 2)

Set arguments equal:
–4x + 2 = 5x + 2

Solve:
–4x = 5x → –9x = 0 → x = 0

Check domain:
Left: –4*0 + 2 = 2 > 0
Right: 5*0 + 2 = 2 > 0 → valid

✔ Answer: x = 0

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15) log₂₀ (–3x – 1) = log₂₀ (–4x – 4)

Set arguments equal:
–3x – 1 = –4x – 4

Solve:
Add 4x: x – 1 = –4 → x = –3

Check domain:
Left: –3*(–3) – 1 = 9 – 1 = 8 > 0
Right: –4*(–3) – 4 = 12 – 4 = 8 > 0 → valid

✔ Answer: x = –3

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16) log (4v + 10) = log (10 – 5v)
(Assuming base 10 unless specified)

Set arguments equal:
4v + 10 = 10 – 5v

Solve:
4v + 5v = 10 – 10 → 9v = 0 → v = 0

Check domain:
Left: 4*0 + 10 = 10 > 0
Right: 10 – 0 = 10 > 0 → valid

✔ Answer: v = 0

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17) log₉ (–3x) – log₉ 10 = log₉ 13

Use quotient rule:
log₉ [ (–3x)/10 ] = log₉ 13

Set arguments equal:
(–3x)/10 = 13 → –3x = 130 → x = –130/3

Check domain: –3x = –3*(–130/3) = 130 > 0 → valid

✔ Answer: x = –130/3

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18) log₉ (–3x) – log₉ 3 = log₉ 35

Quotient rule:
log₉ [ (–3x)/3 ] = log₉ 35 → log₉ (–x) = log₉ 35

So: –x = 35 → x = –35

Check domain: –3x = –3*(–35) = 105 > 0 → valid

✔ Answer: x = –35

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19) log₇ (4x) – log₇ 9 = 1

Quotient rule:
log₇ (4x/9) = 1

Exponential form:
4x/9 = 7¹ = 7 → 4x = 63 → x = 63/4 = 15.75

Check domain: 4x = 63 > 0 → valid

✔ Answer: x = 63/4

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20) log₇ 9 + log₇ (x + 3) = 2

Product rule:
log₇ [9(x + 3)] = 2

Exponential form:
9(x + 3) = 7² = 49 → x + 3 = 49/9 → x = 49/9 – 27/9 = 22/9

Check domain: x + 3 = 22/9 + 27/9 = 49/9 > 0 → valid

✔ Answer: x = 22/9

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21) ln (x – 8) + ln 3 = 5

Product rule:
ln [3(x – 8)] = 5

Exponential form (base e):
3(x – 8) = e⁵ → x – 8 = e⁵ / 3 → x = 8 + e⁵ / 3

We can leave it like this or approximate if needed, but exact is better.

✔ Answer: x = 8 + e⁵ / 3

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22) log (x + 4) + log 3 = 2
(Base 10 assumed)

Product rule:
log [3(x + 4)] = 2

Exponential form:
3(x + 4) = 10² = 100 → x + 4 = 100/3 → x = 100/3 – 12/3 = 88/3

Check domain: x + 4 = 88/3 + 12/3 = 100/3 > 0 → valid

✔ Answer: x = 88/3

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Final Answer:
1) v = 1
2) n = 1/9
3) r = 36
4) x = 125
5) r = –6
6) p = –1/4
7) x = 2401/2
8) k = 131
9) r = –84/5
10) x = 3/7
11) a = 215/2
12) x = 6554/3
13) p = –2
14) x = 0
15) x = –3
16) v = 0
17) x = –130/3
18) x = –35
19) x = 63/4
20) x = 22/9
21) x = 8 + e⁵/3
22) x = 88/3