For how many integers n does the expression √(\log(n^2)-(\log n)^2)/(\log n-3) represent a real number, where \log denotes the base 10 logarithm?

2023 AMC 12B Problem 7 — Solution

2023 AMC 12B Problem 7

Below is the professionally curated solution for Problem 7 of the 2023 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2023 AMC 12B solutions, or check the answer key.

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Concepts:logarithm inequality

Difficulty rating: 1530

For how many integers $n$ does the expression

$\sqrt{\frac{\log(n^2)-(\log n)^2}{\log n-3}}$

represent a real number, where $\log$ denotes the base $10$ logarithm?

$900$

$3$

$902$

$2$

$901$

Solution:

Write $L=\log n.$ Then $\log(n^2)-(\log n)^2=2L-L^2=L(2-L),$ and the fraction is $\dfrac{L(2-L)}{L-3}.$ A sign chart shows this is $\ge 0$ exactly when $L\le 0$ or $2\le L\lt 3.$ Since $n$ is a positive integer, $L\le 0$ forces $n=1,$ while $2\le L\lt 3$ gives $100\le n\le 999,$ which is $900$ values. In total $1+900=901.$

Thus, the correct answer is E.

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