Worksheet for solving logarithmic equations in Pre-Calculus/Trig 3, featuring 12 problems with logarithmic expressions and variables.
Pre-Calculus/Trig 3 worksheet titled "Solving Logarithmic Equations" with 12 problems involving logarithmic equations to solve, including instructions to check for extraneous solutions.
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Step-by-step solution for: SOLVING LOGARITHMIC EQUATIONS | Slides Pre-Calculus | Docsity
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Show Answer Key & Explanations
Step-by-step solution for: SOLVING LOGARITHMIC EQUATIONS | Slides Pre-Calculus | Docsity
Let’s solve each logarithmic equation one by one. Remember: when you have log(A) = log(B), then A must equal B — but only if both A and B are positive (because logs of negative numbers or zero don’t exist). So after solving, we always check that our answer makes the inside of every log positive.
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1.) log(5x) = log(2x + 9)
Set insides equal:
5x = 2x + 9
Subtract 2x:
3x = 9 → x = 3
Check:
log(5*3) = log(15) ✔️
log(2*3 + 9) = log(6+9)=log(15) ✔️
Both positive → valid.
✔ Answer: x = 3
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2.) log(10 – 4x) = log(10 – 3x)
Set insides equal:
10 – 4x = 10 – 3x
Subtract 10 from both sides:
–4x = –3x
Add 4x to both sides:
0 = x → x = 0
Check:
log(10 – 0) = log(10) ✔️
log(10 – 0) = log(10) ✔️
Valid.
✔ Answer: x = 0
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3.) log(4p – 2) = log(–5p + 5)
Set insides equal:
4p – 2 = –5p + 5
Add 5p to both sides:
9p – 2 = 5
Add 2:
9p = 7 → p = 7/9
Check:
Left: 4*(7/9) – 2 = 28/9 – 18/9 = 10/9 > 0 ✔️
Right: –5*(7/9) + 5 = –35/9 + 45/9 = 10/9 > 0 ✔️
Valid.
✔ Answer: p = 7/9
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4.) log(4k – 5) = log(2k – 1)
Set insides equal:
4k – 5 = 2k – 1
Subtract 2k:
2k – 5 = –1
Add 5:
2k = 4 → k = 2
Check:
Left: 4*2 – 5 = 8 – 5 = 3 > 0 ✔️
Right: 2*2 – 1 = 4 – 1 = 3 > 0 ✔️
Valid.
✔ Answer: k = 2
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5.) log(–2a + 9) = log(7 – 4a)
Set insides equal:
–2a + 9 = 7 – 4a
Add 4a to both sides:
2a + 9 = 7
Subtract 9:
2a = –2 → a = –1
Check:
Left: –2*(-1) + 9 = 2 + 9 = 11 > 0 ✔️
Right: 7 – 4*(-1) = 7 + 4 = 11 > 0 ✔️
Valid.
✔ Answer: a = –1
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6.) 2 log₇(–2r) = 0
Divide both sides by 2:
log₇(–2r) = 0
Convert to exponential form:
7⁰ = –2r → 1 = –2r → r = –1/2
Check:
Inside log: –2*(-1/2) = 1 > 0 ✔️
Valid.
✔ Answer: r = –1/2
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7.) –10 + log₃(n + 3) = –10
Add 10 to both sides:
log₃(n + 3) = 0
Convert:
3⁰ = n + 3 → 1 = n + 3 → n = –2
Check:
n + 3 = –2 + 3 = 1 > 0 ✔️
Valid.
✔ Answer: n = –2
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8.) –2 log₅(7x) = 2
Divide both sides by –2:
log₅(7x) = –1
Convert:
5⁻¹ = 7x → 1/5 = 7x → x = 1/(5*7) = 1/35
Check:
7x = 7*(1/35) = 1/5 > 0 ✔️
Valid.
✔ Answer: x = 1/35
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9.) log(–m) + 2 = 4
Subtract 2:
log(–m) = 2
Assume base 10 (since no base written):
10² = –m → 100 = –m → m = –100
Check:
–m = –(-100) = 100 > 0 ✔️
Valid.
✔ Answer: m = –100
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10.) –6 log₃(x – 3) = –24
Divide both sides by –6:
log₃(x – 3) = 4
Convert:
3⁴ = x – 3 → 81 = x – 3 → x = 84
Check:
x – 3 = 84 – 3 = 81 > 0 ✔️
Valid.
✔ Answer: x = 84
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11.) log₁₂(v² + 35) = log₁₂(–12v – 1)
Set insides equal:
v² + 35 = –12v – 1
Bring all terms to left:
v² + 12v + 36 = 0
Factor:
(v + 6)² = 0 → v = –6
Check:
Left: (-6)² + 35 = 36 + 35 = 71 > 0 ✔️
Right: –12*(-6) – 1 = 72 – 1 = 71 > 0 ✔️
Valid.
✔ Answer: v = –6
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12.) log₉(–11x + 2) = log₉(x² + 30)
Set insides equal:
–11x + 2 = x² + 30
Bring all to right:
0 = x² + 11x + 28
Factor:
(x + 4)(x + 7) = 0 → x = –4 or x = –7
Check both:
For x = –4:
Left: –11*(-4) + 2 = 44 + 2 = 46 > 0 ✔️
Right: (-4)² + 30 = 16 + 30 = 46 > 0 ✔️ → Valid
For x = –7:
Left: –11*(-7) + 2 = 77 + 2 = 79 > 0 ✔️
Right: (-7)² + 30 = 49 + 30 = 79 > 0 ✔️ → Also valid? Wait — let me double-check.
Wait — actually, both give positive values inside logs. But let’s plug back into original equation to be sure.
Original: log₉(–11x + 2) = log₉(x² + 30)
At x = –4:
Left: log₉(46), Right: log₉(46) → Equal ✔️
At x = –7:
Left: log₉(79), Right: log₉(79) → Equal ✔️
So both are valid? But wait — did I make an error in setting up?
Wait — let me re-solve the equation:
–11x + 2 = x² + 30
→ 0 = x² + 11x + 28 → yes, factors to (x+4)(x+7)=0 → x=-4, -7
And both make arguments positive → so both are solutions.
But let me check again for x = -7:
–11*(-7) + 2 = 77 + 2 = 79 → OK
x² + 30 = 49 + 30 = 79 → OK
Same for x=-4: 44+2=46, 16+30=46 → OK
So both work.
✔ Answers: x = –4 and x = –7
Wait — but sometimes problems expect only one solution? Let me see if there's any restriction I missed.
No — both satisfy domain and equality. So both are valid.
Final Answer for #12: x = –4, –7
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Now compiling all final answers:
Final Answer:
1. x = 3
2. x = 0
3. p = 7/9
4. k = 2
5. a = –1
6. r = –1/2
7. n = –2
8. x = 1/35
9. m = –100
10. x = 84
11. v = –6
12. x = –4, –7
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1.) log(5x) = log(2x + 9)
Set insides equal:
5x = 2x + 9
Subtract 2x:
3x = 9 → x = 3
Check:
log(5*3) = log(15) ✔️
log(2*3 + 9) = log(6+9)=log(15) ✔️
Both positive → valid.
✔ Answer: x = 3
---
2.) log(10 – 4x) = log(10 – 3x)
Set insides equal:
10 – 4x = 10 – 3x
Subtract 10 from both sides:
–4x = –3x
Add 4x to both sides:
0 = x → x = 0
Check:
log(10 – 0) = log(10) ✔️
log(10 – 0) = log(10) ✔️
Valid.
✔ Answer: x = 0
---
3.) log(4p – 2) = log(–5p + 5)
Set insides equal:
4p – 2 = –5p + 5
Add 5p to both sides:
9p – 2 = 5
Add 2:
9p = 7 → p = 7/9
Check:
Left: 4*(7/9) – 2 = 28/9 – 18/9 = 10/9 > 0 ✔️
Right: –5*(7/9) + 5 = –35/9 + 45/9 = 10/9 > 0 ✔️
Valid.
✔ Answer: p = 7/9
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4.) log(4k – 5) = log(2k – 1)
Set insides equal:
4k – 5 = 2k – 1
Subtract 2k:
2k – 5 = –1
Add 5:
2k = 4 → k = 2
Check:
Left: 4*2 – 5 = 8 – 5 = 3 > 0 ✔️
Right: 2*2 – 1 = 4 – 1 = 3 > 0 ✔️
Valid.
✔ Answer: k = 2
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5.) log(–2a + 9) = log(7 – 4a)
Set insides equal:
–2a + 9 = 7 – 4a
Add 4a to both sides:
2a + 9 = 7
Subtract 9:
2a = –2 → a = –1
Check:
Left: –2*(-1) + 9 = 2 + 9 = 11 > 0 ✔️
Right: 7 – 4*(-1) = 7 + 4 = 11 > 0 ✔️
Valid.
✔ Answer: a = –1
---
6.) 2 log₇(–2r) = 0
Divide both sides by 2:
log₇(–2r) = 0
Convert to exponential form:
7⁰ = –2r → 1 = –2r → r = –1/2
Check:
Inside log: –2*(-1/2) = 1 > 0 ✔️
Valid.
✔ Answer: r = –1/2
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7.) –10 + log₃(n + 3) = –10
Add 10 to both sides:
log₃(n + 3) = 0
Convert:
3⁰ = n + 3 → 1 = n + 3 → n = –2
Check:
n + 3 = –2 + 3 = 1 > 0 ✔️
Valid.
✔ Answer: n = –2
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8.) –2 log₅(7x) = 2
Divide both sides by –2:
log₅(7x) = –1
Convert:
5⁻¹ = 7x → 1/5 = 7x → x = 1/(5*7) = 1/35
Check:
7x = 7*(1/35) = 1/5 > 0 ✔️
Valid.
✔ Answer: x = 1/35
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9.) log(–m) + 2 = 4
Subtract 2:
log(–m) = 2
Assume base 10 (since no base written):
10² = –m → 100 = –m → m = –100
Check:
–m = –(-100) = 100 > 0 ✔️
Valid.
✔ Answer: m = –100
---
10.) –6 log₃(x – 3) = –24
Divide both sides by –6:
log₃(x – 3) = 4
Convert:
3⁴ = x – 3 → 81 = x – 3 → x = 84
Check:
x – 3 = 84 – 3 = 81 > 0 ✔️
Valid.
✔ Answer: x = 84
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11.) log₁₂(v² + 35) = log₁₂(–12v – 1)
Set insides equal:
v² + 35 = –12v – 1
Bring all terms to left:
v² + 12v + 36 = 0
Factor:
(v + 6)² = 0 → v = –6
Check:
Left: (-6)² + 35 = 36 + 35 = 71 > 0 ✔️
Right: –12*(-6) – 1 = 72 – 1 = 71 > 0 ✔️
Valid.
✔ Answer: v = –6
---
12.) log₉(–11x + 2) = log₉(x² + 30)
Set insides equal:
–11x + 2 = x² + 30
Bring all to right:
0 = x² + 11x + 28
Factor:
(x + 4)(x + 7) = 0 → x = –4 or x = –7
Check both:
For x = –4:
Left: –11*(-4) + 2 = 44 + 2 = 46 > 0 ✔️
Right: (-4)² + 30 = 16 + 30 = 46 > 0 ✔️ → Valid
For x = –7:
Left: –11*(-7) + 2 = 77 + 2 = 79 > 0 ✔️
Right: (-7)² + 30 = 49 + 30 = 79 > 0 ✔️ → Also valid? Wait — let me double-check.
Wait — actually, both give positive values inside logs. But let’s plug back into original equation to be sure.
Original: log₉(–11x + 2) = log₉(x² + 30)
At x = –4:
Left: log₉(46), Right: log₉(46) → Equal ✔️
At x = –7:
Left: log₉(79), Right: log₉(79) → Equal ✔️
So both are valid? But wait — did I make an error in setting up?
Wait — let me re-solve the equation:
–11x + 2 = x² + 30
→ 0 = x² + 11x + 28 → yes, factors to (x+4)(x+7)=0 → x=-4, -7
And both make arguments positive → so both are solutions.
But let me check again for x = -7:
–11*(-7) + 2 = 77 + 2 = 79 → OK
x² + 30 = 49 + 30 = 79 → OK
Same for x=-4: 44+2=46, 16+30=46 → OK
So both work.
✔ Answers: x = –4 and x = –7
Wait — but sometimes problems expect only one solution? Let me see if there's any restriction I missed.
No — both satisfy domain and equality. So both are valid.
Final Answer for #12: x = –4, –7
---
Now compiling all final answers:
Final Answer:
1. x = 3
2. x = 0
3. p = 7/9
4. k = 2
5. a = –1
6. r = –1/2
7. n = –2
8. x = 1/35
9. m = –100
10. x = 84
11. v = –6
12. x = –4, –7
Parent Tip: Review the logic above to help your child master the concept of exponential logarithmic equations worksheet.
