Question

What is ${\log }_{10}\left(\frac{3}{2}\right)+{\log }_{10}\left(\frac{4}{3}\right)+{\log }_{10}\left(\frac{5}{4}\right)+....$ up to $8$ terms equal to?

• A. $0$
• B. $1$
• C. ${\log }_{10}5$
• D. None of the above

Answer 1

Answer: (C)

${\log }_{10}5$

We need to find sum of ${\log }_{10}\left(\frac{3}{2}\right)+{\log }_{10}\frac{4}{3}+{\log }_{10}\frac{5}{4}+....$ upto $8$ terms

As their bases are equal i.e. $10$, the terms can be multiplied.

$\text{Sum}={\log }_{10}\frac{3}{2}\times \frac{4}{3}\times \frac{5}{4}\times \frac{6}{5}\times \frac{7}{6}\times \frac{8}{7}\times \frac{9}{8}\times \frac{10}{9}$
$={\log }_{10}5$