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

Original Equation:

The equation provided is:

\(\log_{\sqrt{3}}(x) + \frac{\log_x(25)}{\log_x(0.008)} = \frac{16}{3}\)

Step 1: Simplify Logarithmic Terms

The equation is transformed to:

\(⇒ 2 \log_3(x) + \log_{0.008}(25) = \frac{16}{3}\)

Step 2: Express \(\log_{0.008}(25)\)

The second logarithmic term is expressed using logarithm properties:

\(⇒ 2 \log_3(x) + \log_{\frac{8}{1000}}(25) = \frac{16}{3}\)

Step 3: Further Simplification

It is known that:

\(log_{\frac{8}{1000}} 25 = \log_5(25)^{\left(-3\right)} = \frac{2}{3}\)

Substituting this value yields:

\(⇒ 2 \log_3(x) - \frac{2}{3} = \frac{16}{3}\)

Step 4: Solve for \(\log_3(x)\)

Adding \(\frac{2}{3}\) to both sides:

\(⇒ 2 \log_3(x) = \frac{16}{3} + \frac{2}{3} = 6\)

Step 5: Solve for \(x\)

Dividing both sides by 2:

\(⇒ \log_3(x^2) = 6\)

This implies:

\(⇒ x^2 = 3^6\)

Step 6: Final Calculation

The final expression is:

\(log_3(3 \cdot x^2) = log_3(3 \cdot 3^6) = log_3(3^7) = 7\)

Conclusion:

Therefore, the correct answer is (C): 7.