ACT Math Practice Question #1233 (Hard (35)) | Test Citadel
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ACT Math Difficulty: Hard (35)

Digital ACT Math Practice Question #1233

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If $\log_b(3) = x$ and $\log_b(5) = y$, what is the value of $\log_b(45\sqrt{b})$ expressed in terms of $x$ and $y$?
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Decompose $45\sqrt{b} = 3^2 \cdot 5^1 \cdot b^{1/2} \implies 2x + 1y + 1/2$.

Masterclass Solution & Distractor Trap Analysis

Notice $45 = 9 \times 5 = 3^2 \times 5$. Using log product and power rules: $$\log_b(45\sqrt{b}) = \log_b(3^2 \cdot 5 \cdot b^{1/2}) = \log_b(3^2) + \log_b(5) + \log_b(b^{1/2})$$ $$= 2\log_b(3) + \log_b(5) + \frac{1}{2}\log_b(b) = 2x + y + ...

Distractor Analysis: Trap choice eliminates careless test-takers who confuse roots with coordinates...

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