答案錯誤

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12. 阿淞為了方便記住自己設定的四位數密碼，隨身攜帶一張紙條，上面註記： \left\ulcorner a=25 \frac{2}{\log S}\right. $\left(\log _{2} 81 \cdot \log _{3} 1024\right)$ 」，其中 $a$ 即為四位數之密碼，請問密碼為 $\qquad$
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Answer from Sia

Posted 23 days ago

Solution by Steps

step 1

We start with the expression $a = \frac{25^2}{\log_5(1)} + \log_2(81) \cdot \log_3(1024)$

step 2

Since $\log_5(1) = 0$, the first term becomes undefined, so we focus on $\log_2(81) \cdot \log_3(1024)$

step 3

Using the change of base formula, we can express this as $\frac{\log(81)}{\log(2)} \cdot \frac{\log(1024)}{\log(3)}$

step 4

We know that $\log(81) = 4 \log(3)$ and $\log(1024) = 10 \log(2)$, substituting gives us $\frac{4 \log(3) \cdot 10 \log(2)}{\log(2) \log(3)} = 40$

Answer

The four-digit password $a$ is approximately $40$

Key Concept

Understanding logarithmic properties and change of base formula

Explanation

The calculation shows how to simplify logarithmic expressions to find the value of $a$ effectively.

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