Question 5.

It is given that $\log_{\sqrt{3}}(x) + \frac{\log_3(25)}{\log_3(0.008)} = \frac{16}{3}$, which can be written as:

=> $2 \log_3 x + \log_{0.008} 25 = \frac{16}{3}$

=> $2 \log_3 x + \log_{\frac{8}{1000}} 25 = \frac{16}{3}$

=> $2 \log_3 x + \log_{\frac{1}{125}} 25 = \frac{16}{3}$

=> $2 \log_3 x + \log_{5^{-3}} (5)^2 = \frac{16}{3}$

=> $2 \log_3 x - \frac{2}{3} = \frac{16}{3}$

=> $2 \log_3 x = \frac{16}{3} + \frac{2}{3}$

=> $2 \log_3 x = 6$

=> $\log_3 x^2 = 6 \Rightarrow x^2 = 3^6$

Hence, $\log_3 \left(3 \cdot x^2\right) = \log_3 \left(3 \cdot 3^6\right) = \log_3 3^7 = 7$