Correct Answer - Option 4 : 2
Concept:
- \(\rm log_bm.log_ab = log_am\)
- \(\rm log_ba =\frac{1}{ log_ab}\)
Calculation:
Here we have to find the value of \(\rm \frac{log_{7}3 \times log_{2}7\times log_{3}2}{log_{5}\sqrt{15}}+\frac{1}{{log_{3}\sqrt{15}}} \)
⇒ \(\rm \rm \frac{log_{7}3 \times log_{2}7\times log_{3}2}{log_{5}\sqrt{15}}+\frac{1}{{log_{3}\sqrt{15}}} = \frac{log_{7}3 \times log_{3}7}{log_{5}\sqrt{15}}+\frac{1}{{log_{3}\sqrt{15}}}\) ( ∴ \(\rm log_bm \times log_ab = log_am\))
⇒ \(\rm \frac{log_{7}3 \times log_{3}7}{log_{5}\sqrt{15}}+\frac{1}{{log_{3}\sqrt{15}}}= \frac{log_{3}3}{log_{5}\sqrt{15}}+\frac{1}{{log_{3}\sqrt{15}}}\) ( ∴ \(\rm log_bm.log_ab = log_am\))
= \(\rm \frac{1}{log_{5}\sqrt{15}}+\frac{1}{{log_{3}\sqrt{15}}}\)
= \(\rm {log_{\sqrt{15}}5}+{{log_{\sqrt{15}}3}}\) ( ∴ \(\rm log_ba =\frac{1}{ log_ab}\))
= \(\rm {log_{\sqrt{15}}15}\)
= \(\rm {log_{\sqrt{15}}{(\sqrt{15})}^2}\)
= \(\rm 2 {log_{\sqrt{15}}{(\sqrt{15})}}\)
= 2
Hence, option D is correct.