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2.13 · Steps: Single Log/Exp
What are the 4 steps to solve an equation with a single log or exponential?
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2.13 · Two Target Forms
When condensing multiple logs to solve, what are the two target forms?
Form 1: log_b(f) = c • Form 2: log_b(f) = log_b(g)
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2.13 · Solve: Single Exp
Solve for x (exact): 4หฃ − 2 = 7
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2.13 · Solve: Single Log
Solve: 2 log₃(x+3) + 4 = 10
Isolate the log first, then convert
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2.13 · Extraneous Solutions
Solve: log₃(x+2) + log₃(x−3) = log₃14. Are both solutions valid?
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2.13 · Common Base
Solve: 9^(2x) = 27^(x+2)
9 = 3² and 27 = 3³
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2.13 · Find Inverse
Find f⁻¹(x) for f(x) = 3(2)^(x+4) − 15
Switch x↔y, isolate the exponential, convert to log
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2.13 · Find Inverse
Find g⁻¹(x) for g(x) = −3 log₅(x−4) + 6
Switch x↔y, isolate the log, convert to exponential
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✓ Two Target Forms
Form 1: log_b(f) = c → f = b^c Form 2: log_b(f) = log_b(g) → f = g
Form 1: convert to exponential and solve. Form 2: set arguments equal and solve. Always check for extraneous solutions!
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✓ Steps: Single Log/Exp
1. Isolate 2. Convert form 3. Solve 4. Simplify (exact)
Example: 4ˣ−2=7 → 4ˣ=9 → x=log₄9. Example: 2log₃(x+3)+4=10 → log₃(x+3)=3 → x+3=27 → x=24.
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✓ Solve: Single Log
x = 24
2log₃(x+3)=6 → log₃(x+3)=3 → x+3=3³=27 → x=24.
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✓ Solve: Single Exp
x = log₄ 9
4ˣ=9. Convert to log form: x = log₄ 9. (Use change of base to get โ 1.585 on a calculator.)
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✓ Common Base
x = 6
(3²)^(2x) = (3³)^(x+2) → 3^(4x) = 3^(3x+6) → 4x = 3x+6 → x=6.
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✓ Extraneous Solutions
x = 5 only. x = −4 is extraneous.
(x+2)(x−3)=14 → x²−x−20=0 → x=5 or x=−4. x=−4 fails domain (x−3=−7<0). Only x=5 ✓.
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✓ Inverse of g(x)
g⁻¹(x) = 5^((6−x)/3) + 4
Switch: x=−3log₅(y−4)+6. Rearrange: (6−x)/3 = log₅(y−4). Convert: 5^((6−x)/3) = y−4. Add 4.
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✓ Inverse of f(x)
f⁻¹(x) = log₂((x+15)/3) − 4
Switch, add 15, divide by 3: (x+15)/3 = 2^(y+4). Convert to log: log₂((x+15)/3) = y+4. Subtract 4.
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2.13 · No Solution Check
Solve: log₄(x) − 3log₄2 = log₄(x+7). Is there a solution?
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2.13 · Exponential Inequality
Solve: 1 − 4(2)^(x−3) ≤ −31
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2.13 · Multiple Logs โ Quadratic
ln(x+10) − ln(x) = ln(x+3). Solve and state any domain restriction.
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2.13 · FRQ Tip
On AP FRQ #4 (no calculator), what form should you leave answers in?
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✓ Exponential Inequality
x ≥ 6
−4(2)^(x−3) ≤ −32 → (2)^(x−3) ≥ 8=2³ → x−3 ≥ 3 → x ≥ 6.
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✓ No Solution
No solution
log₄(x/8) = log₄(x+7) → x/8 = x+7 → x=−8. But domain requires x > 0. So no solution.
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✓ FRQ Tip
Exact form: e.g. ln(5/2), log₄9, 3e²
Never give decimals on FRQ #4. Rewrite step-by-step using log/exp properties. Show all work โ partial credit counts!
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✓ Multiple Logs → Quadratic
x = (−2+√44)/2 ≈ 2.317
(x+10)/x = x+3 → x²+2x−10=0 → x=(−2±√44)/2. Negative root rejected (domain x>0).