Question

Which of the following when simplified, vanishes

• A. (A) $\frac{1}{\log_3 2} + \frac{2}{\log_9 4} - \frac{3}{\log_{27} 8}$
• B. (B) $\log_2 (\frac{2}{3}) + \log_4 (\frac{9}{4})$
• C. (C) – $\log_8 \log_4 \log_2 16$
• D. (D) $\log_{10} \cot 1^\circ + \log_{10} \cot 2^\circ + \log_{10} \cot 3^\circ + \dots + \log_{10} \cot 89^\circ$

Answer 1

Answer: (A), (B), (C), (D)

(A), (B), (C), (D)

Answer 2

(A) $\frac{1}{\log_3 2} + \frac{2}{\log_9 4} - \frac{3}{\log_{27} 8} = \log_2 3 + 2\log_2 3 - 3\log_2 3 = 0$.

(B) $\log_2 (\frac{2}{3}) + \log_4 (\frac{9}{4}) = (\log_2 2 - \log_2 3) + (\log_4 9 - \log_4 4) = (1 - \log_2 3) + (\log_2 3 - 1) = 0$.

(C) – $\log_8 \log_4 \log_2 16 = -\log_8 \log_4 4 = -\log_8 1 = 0$.

(D) $\log_{10} (\cot 1^\circ \cdot \cot 2^\circ \cdot \dots \cdot \cot 89^\circ)$. Since $\cot(90^\circ - x) = \tan x$, the product is $(\cot 1^\circ \tan 1^\circ) \dots (\cot 44^\circ \tan 44^\circ) \cot 45^\circ = 1 \cdot 1 \cdot \dots \cdot 1 \cdot 1 = 1$. Thus, $\log_{10} 1 = 0$.