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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∻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. ⚹◊What is the value of $\log_2 81$ and how does it contribute to solving for $a$?⍭ Generate me a similar question◊

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