AMC12 2023 B

AMC12 2023 B · Q7

AMC12 2023 B · Q7. It mainly tests Linear inequalities, Manipulating equations.

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?

有且仅有有多少个整数$n$，使得表达式\[\sqrt{\frac{\log (n^2) - (\log n)^2}{\log n - 3}}\]表示一个实数，其中log表示以10为底的对数？

(A) 900 900

(B) 3 3

(D) 2 2

(E) 901 901

Answer

Correct choice: (E)

正确答案：（E）

Solution

We have \begin{align*} \sqrt{\frac{\log \left( n^2 \right) - \left( \log n \right)^2}{\log n - 3}} & = \sqrt{\frac{2 \log n - \left( \log n \right)^2}{\log n - 3}} \\ & = \sqrt{\frac{\left( \log n \right) \left( 2 - \log n\right)}{\log n - 3}} . \end{align*} Because $n$ is an integer and $\log n$ is well defined, $n$ must be a positive integer. Case 1: $n = 1$ or $10^2$. The above expression is 0. So these are valid solutions. Case 2: $n \neq 1, 10^2$. Thus, $\log n > 0$ and $2 - \log n \neq 0$. To make the above expression real, we must have $2 < \log n < 3$. Thus, $100 < n < 1000$. Thus, $101 \leq n \leq 999$. Hence, the number of solutions in this case is 899. Putting all cases together, the total number of solutions is $\boxed{\textbf{(E) 901}}$.

我们有 \begin{align*} \sqrt{\frac{\log \left( n^2 \right) - \left( \log n \right)^2}{\log n - 3}} & = \sqrt{\frac{2 \log n - \left( \log n \right)^2}{\log n - 3}} \\ & = \sqrt{\frac{\left( \log n \right) \left( 2 - \log n\right)}{\log n - 3}} . \end{align*} 因为$n$是整数且$\log n$有定义，所以$n$必须是正整数。情况1: $n = 1$ 或 $10^2$。上述表达式为0。所以这些是有效解。情况2: $n \neq 1, 10^2$。因此，$\log n > 0$ 且 $2 - \log n \neq 0$。要使上述表达式为实数，必须有$2 < \log n < 3$。因此，$100 < n < 1000$。因此，$101 \leq n \leq 999$。故此情况解的个数为899。将所有情况综合，总解的个数为 $\boxed{\textbf{(E) 901}}$。

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