Question

Which of the following when simplified, vanishes

• A. $\frac{1}{\log_3 2} + \frac{2}{\log_9 4} - \frac{3}{\log_{27} 8}$
• B. $\log_2 (\frac{2}{3}) + \log_4 (\frac{9}{4})$
• C. – log log4 log2 16
• D. $\log_{10} \cot 1^\circ + \log_{10} \cot 2^\circ + \log_{10} \cot 3^\circ + ......... + \log_{10} \cot 89^\circ$

Answer 1

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

A, B, C, D

Answer 2

Option (A): Simplifies to $\log_2 3 + 2\log_2 3 - 3\log_2 3 = 0$.

Option (B): Simplifies to $(1 - \log_2 3) + (\log_2 3 - 1) = 0$.

Option (C): Evaluates to $-\log(\log_4(4)) = -\log(1) = 0$.

Option (D): The sum of logs is the log of the product. The product $\cot 1^\circ \cdot \cot 2^\circ \cdots \cot 89^\circ = 1$. Thus, $\log_{10}(1) = 0$. All options vanish.