AMC12 2024 B

AMC12 2024 B · Q8

AMC12 2024 B · Q8. It mainly tests Logarithms (rare), Manipulating equations.

What value of $x$ satisfies \[\frac{\log_2x \cdot \log_3x}{\log_2x+\log_3x}=2?\]

什么$x$满足 \[\frac{\log_2x \cdot \log_3x}{\log_2x+\log_3x}=2?\]

(A) 25 25

(B) 32 32

(D) 42 42

(E) 48 48

Answer

Correct choice: (C)

正确答案：（C）

Solution

We have log2⁡x⋅log3⁡x=2(log2⁡x+log3⁡x)1=2(log2⁡x+log3⁡x)log2⁡x⋅log3⁡x1=2(1log3⁡x+1log2⁡x)1=2(logx⁡3+logx⁡2)logx⁡6=12x12=6x=36 so $\boxed{\textbf{(C) }36}$

我们有 $\log_2 x \cdot \log_3 x = 2(\log_2 x + \log_3 x)$ $\frac{1}{\log_2 x \cdot \log_3 x} = 2\left(\frac{1}{\log_3 x} + \frac{1}{\log_2 x}\right)$ $\frac{1}{\log_2 x \cdot \log_3 x} = 2(\log_x 3 + \log_x 2)$ $\log_x 6 = \frac{1}{2}$ $x^{1/2} = 6$ $x = 36$ 所以$\boxed{\textbf{(C) }36}$

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