1954 AMC 12 Problem 15

Attempt Problem 15 of the 1954 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1954 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

15.

log⁡125\log125 equals:
100log⁡1.25100\log1.25
5log⁡35\log3
3log⁡253\log25
3−3log⁡23-3\log2
(log⁡25)(log⁡5)(\log25)(\log5)
Answer: D
Concepts:logarithmalgebraic manipulation
Small hint:
Write 125=10008125=\frac{1000}{8}
Big hint:
Use log⁡1000=3\log1000=3 and log⁡8=3log⁡2\log8=3\log2
Solution:
Using logarithm laws, log⁡125=log⁡(10008)=3−log⁡(23)=3−3log⁡2. \begin{aligned} \log125 &=\log\left(\frac{1000}{8}\right)\\ &=3-\log(2^3)\\ &=3-3\log2. \end{aligned}
Thus, the correct answer is D.
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