Question

If ${\log }_{2x}20\sqrt{6}={\log }_{3x}90$, then characteristic of ${\log }_5{x}^3$ is equal to

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (C)

$\text{ Let }{\log }_{2x}20\sqrt{6}={\log }_{3x}90=k$ $\Rightarrow 20\sqrt{6}=(2x{)}^k\text{ and }90=(3x{)}^k$ $\therefore \frac{90}{20\sqrt{6}}={\left(\frac{3}{2}\right)}^k\Rightarrow {\left(\frac{3}{2}\right)}^{3/2}={\left(\frac{3}{2}\right)}^k$ $\therefore 20\sqrt{6}=(2x{)}^{3/2}\Rightarrow 400\times 6=8x^3\Rightarrow x^3=300$ $\therefore \text{ Characteristic of }{\log }_5{x}^3=3$