Question

If ${\log }_{10}2=a$ and ${\log }_{10}3=b$, then $\log 3\frac{1}{8}$ can be expressed as

• A. $2-a$
• B. $2-5a$
• C. $2+5a$
• D. $2+a$

Answer 1

Answer: (B)

$2-5a$

Let, $x=\log 3\frac{1}{8}$ = $\log \frac{25}{8}$ = $\log \frac{5^2}{2^3}$

$\Rightarrow x=\log 5^2-\log 2^3$ $(\because \log \frac{a}{b}=\log a-logb)$
$\Rightarrow x=2\log 5-3\log 2$ $(\because \log a^b=b\log a)$
$\Rightarrow x=2(\log 10-\log 2)-3\log 2$
$\Rightarrow x=2-2a-3a$ ($\because \log 10=1$)
$\Rightarrow x=2-5a$