Question

If ${\log }_{x}2=a,{\log }_{x}3=b,{\log }_{x}5=c,$ then find the value of the following in terms of a, b, and c,
(i) ${\log }_{x}16$ (ii) ${\log }_{x}75$ (iii) ${\log }_{x}60$

• A. $4a,2a+b,2a+b+c$
• B. $4a,2c+b,2a+b+c$
• C. $4a,2c+b,2c+b+a$
• D. $4a,2c+b,2a+3b+c$

Answer 1

The correct option is B $4a,2c+b,2a+b+c$

We know that
$\log AB=\log A+\log B$
Also, $\log a^b=b\log a$
So,
(i) ${\log }_{x}16={\log }_x{2}^4=4{\log }_{x}2=4a$
(ii) ${\log }_{x}75={\log }_x(3\times 5^2)={\log }_{x}3+2{\log }_{x}5=b+2c$
(iii) ${\log }_{x}60={\log }_x(2^2\times 3\times 5)=2{\log }_{x}2+{\log }_{x}3+{\log }_{x}5=2a+b+c$