Correct Answer - Option 1 : -4
Concept:
- \( \rm log_aa=1\)
- \(log_{a^b}x=\frac{1}{b}log_ax\)
Calculation:
Here, we have to find the value of the expression \(\rm log_{2}[log_{0.5}{(log_4(log_{3}9))}] + log_{0.5^{0.25}}2\)
⇒ \(\rm log_{2}[log_{0.5}{(log_4(log_{3}9))}] + log_{0.5^{0.25}}2= \rm log_{2}[log_{0.5}{(log_4(log_{3}3 ^2))}] + log_{0.5^{0.25}}2\)
= \(\rm log_{2}[log_{0.5}{(log_4(2log_{3}3 ))}] + log_{0.5^{0.25}}2\)
= \(\rm log_{2}[log_{0.5}{(log_42 )}] + log_{0.5^{0.25}}2\)
= \(\rm log_{2}[log_{0.5}{(log_{2^2}2 )}] + log_{0.5^{0.25}}2\)
= \(\rm log_{2}[log_{0.5}{(\frac{1}{2}log_{2}2 )}] +\frac{1}{0.25} log_{0.5}2\)
= \(\rm log_{2}[log_{0.5}{(0.5)}] +4 log_{0.5}2\)
= \(\rm log_{2}1 +4 log_{2^{-1}}2\)
= \(\rm 0 +\frac{4}{-1} log_{2}2\)
= \({-4}\ log_{2}2 = - 4\)
Hence, option A is correct.
