log 4 + 1/3 log 125 − 1/5 log 32
2025
log 4 + 1/3 log 125 − 1/5 log 32
Answer: A. 1 — Three logarithm identities are used here: the power rule log(an) = n·log a, the product rule log a + log b = log(a×b), and the quotient rule log a − log b =…
- A.
1
- B.
4
- C.
3
- D.
6
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Correct answer: A
Three logarithm identities are used here: the power rule log(an) = n·log a, the product rule log a + log b = log(a×b), and the quotient rule log a − log b = log(a/b).
Rewrite the non-integer arguments as clean powers: 125 = 53 and 32 = 25.
Apply the power rule to each term: (1/3) log 125 = (1/3) log(53) = log 5, and (1/5) log 32 = (1/5) log(25) = log 2.
Substitute these back into the expression: log 4 + (1/3) log 125 − (1/5) log 32 becomes log 4 + log 5 − log 2.
Combine the first two terms with the product rule: log 4 + log 5 = log(4 × 5) = log 20.
Combine with the quotient rule: log 20 − log 2 = log(20/2) = log 10.
Since the logarithm is base 10, log 10 = 1, so the expression equals 1.
Cross-check with a different grouping: writing 4 = 22 gives log 4 = 2 log 2, so the full expression is 2 log 2 + log 5 − log 2 = log 2 + log 5 = log(2 × 5) = log 10 = 1, the same result.