Question

For some positive real number \(x\) , if  \(log_{\sqrt 3}(x)+\frac{log_x(25)}{log_x(0.008)}=\frac{16}{3}\), then the value of \(log_3(3x^2)\) is 

• A. 4
• B. 6
• C. 7
• D. 9

Answer 1

The Correct Option is C

Given : \(\log_{\sqrt3}(x)+\frac{\log_x(25)}{\log_x(0.008)}=\frac{16}{3}\), which can be stated as :

⇒ 2 log₃x + log_0.0825 = \(\frac{16}{3}\)

⇒ 2 log₃x + \(\log_{\frac{8}{1000}}\)25 = \(\frac{16}{3}\)

⇒ 2 log₃x + \(\log_{5^{-3}}(5)^2=\frac{16}{3}\)

⇒ 2 log₃x - \(\frac{2}{3}=\frac{16}{3}\)

⇒ 2 log₃x = \(\frac{16}{3}+\frac{2}{3}\)
⇒ 2 log₃x = 6
⇒  log₃x² = 6
⇒ x² = 3⁶
Therefore, log₃ (3.x²)
= log₃ (3.3⁶)
= log₃ 3⁷
= 7

So, the correct option is (C): 7.