Question

If $\log _3 2=x$, then the value of $\frac{\log _{10} 72}{\log _{10} 24}$ is:

• A. $\frac{1+x}{1-x}$
• B. $\frac{2+3 x}{1+3 x}$
• C. $\frac{2-3 x}{2+3 x}$
• D. $\frac{3 x+1}{3 x+2}$

Answer 1

Answer: (B)

Given, $\log _3 2=x$
$\frac{\log _{10} 72}{\log _{10} 24}=\log _{24} 72=\log _{24}(24 \times 3)$
$=\log _{24} 24+\log _{24} 3=1+\frac{\log 3}{\log 24} $
$=1+\frac{\log _3 3}{\log _3 24} $
$=1+\frac{1}{\log _3(3 \times 8)} $
$=1+\frac{1}{\log _3 3+\log _3 8} $
$=1+\frac{1}{1+\log _3 2^3} $
$=1+\frac{1}{1+3 \log _3 2} $
$=\frac{1+3 \log _3 2+1}{1+3 \log _3 2}=\frac{2+3 x}{1+3 x} .$