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Flip to reveal answersYou see 2 log x + log y − log z. Which way do the log laws take you, and to what?
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All 10 Flashcards — Laws of logarithms
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Question
You see 2 log x + log y − log z. Which way do the log laws take you, and to what?
Answer
Coefficients go UP as powers, + becomes ×, − becomes ÷: 2 log x + log y − log z = log(x²y/z). To go the other way (one log → several), read the laws right-to-left. Spot whether they want 'a single log' or 'expanded'.
Question
Can you split log(x + y)?
Answer
No — that is the #1 trap. The laws only act on products, quotients and powers, never on a sum.
Question
What are log_a 1 and log_a a?
Answer
log_a 1 = 0 (since a⁰ = 1) and log_a a = 1 (since a¹ = a).
Question
The power law — what does it do?
Answer
It brings an exponent down to the front as a coefficient: log_a(xᵐ) = m log_a x. Example: log 8 = log 2³ = 3 log 2.
Question
Write ln 6 + 2 ln 3 − ln 2 as a single logarithm.
Answer
2 ln 3 = ln 9; then ln 6 + ln 9 − ln 2 = ln(6 × 9 ÷ 2) = ln 27.
Question
Given log 2 = p and log 3 = q, write log 24 in terms of p and q.
Answer
24 = 2³ × 3, so log 24 = 3 log 2 + log 3 = 3p + q.
Question
Your calculator only does log₁₀ and ln, but you need log₂ 50. What do you do?
Answer
Change of base: log₂ 50 = (log 50)/(log 2) ≈ 5.64 (any base b works: log_a x = (log_b x)/(log_b a)). Use it whenever the base is not 10 or e, or to combine logs of different bases.
Question
Evaluate log₈ 32.
Answer
Change to base 2: log₂32 ÷ log₂8 = 5 ÷ 3 = 5/3.
Question
Given log 2 = p and log 3 = q (base 10), write log₃ 8 in terms of p and q.
Answer
Change to base 10: log 8 ÷ log 3 = 3 log 2 ÷ log 3 = 3p/q.
Question
Expand log₂(8x³).
Answer
Product then power law: log₂8 + log₂x³ = 3 + 3 log₂ x.
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Full study notes for Laws of logarithms
Topic 1.7 hub
Exponent & log laws
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