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Q.
If $\log _{2 x} 20 \sqrt{6}=\log _{3 x} 90$, then characteristic of $\log _5 x^3$ is equal to
Continuity and Differentiability
Solution:
$ \text { Let } \log _{2 x } 20 \sqrt{6}=\log _{3 x } 90= k $
$\Rightarrow 20 \sqrt{6}=(2 x )^{ k } \text { and } 90=(3 x )^{ 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}=(2 x )^{3 / 2} \Rightarrow 400 \times 6=8 x ^3 \Rightarrow x ^3=300 $
$\therefore \text { Characteristic of } \log _5 x ^3=3 $