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Flip to reveal answersHow do you solve aˣ = b when the bases can be matched?
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All 10 Flashcards — Exponential & log equations
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Question
How do you solve aˣ = b when the bases can be matched?
Answer
Write both sides as powers of the same base, then equate the exponents. E.g. 4ˣ = 8 → 2²ˣ = 2³ → 2x = 3 → x = 3/2.
Question
How do you solve aˣ = b when the bases will not match?
Answer
Take logs of both sides; the power law brings x down: x log a = log b, so x = log b / log a.
Question
Why does taking logs solve an exponential equation?
Answer
The power law: log aˣ = x log a turns the unknown exponent into a coefficient you can divide out.
Question
Solve 5ˣ = 20 exactly.
Answer
x log 5 = log 20, so x = log 20 / log 5 = log₅ 20 (≈ 1.86).
Question
How do you solve a log equation like log_a(expr) = c?
Answer
Convert to exponential form: expr = aᶜ, then solve. E.g. log₃(x − 1) = 2 → x − 1 = 9 → x = 10.
Question
Solve ln(x² − 16) = 0.
Answer
e⁰ = x² − 16, so x² = 17 and x = ±√17 (both keep the argument positive).
Question
Two logarithms in one equation — what is the first step?
Answer
Combine them into a single log with the product or quotient law, then convert to exponential form and solve.
Question
Why must you check solutions of a log equation?
Answer
A logarithm needs a positive argument, so reject any solution that makes an argument ≤ 0.
Question
Given log_k 81 = 4, find the base k.
Answer
k⁴ = 81, so k = ⁴√81 = 3 (take the positive root).
Question
On Paper 2, how do you solve an exponential equation graphically?
Answer
Enter each side as Y₁ and Y₂, graph, then use 2nd → TRACE → 5: intersect. Check for more than one crossing.
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Full study notes for Exponential & log equations
Topic 1.7 hub
Exponent & log laws
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