Question

If $x=1+\log 2-\log 5$, $y=2\log 3$, $z=\log a-\log 5$ and also $x+y=2z$, then the value of $a$ is equal to

• A. $16$
• B. $20$
• C. $12$
• D. $30$

Answer 1

Answer: (D)

$30$

$\text{Given},x=1+\log 2-\log 5\text{Assuming base 10}x={\log }_{10}10+{\log }_{10}2-{\log }_{10}5={\log }_{10}\frac{20}{5}={\log }_{10}4\text{Also,}y=2{\log }_{10}3={\log }_{10}9\text{\&}z={\log }_{10}a-{\log }_{10}5={\log }_{10}\frac{a}{5}\text{Now,}x+y=2z\therefore {\log }_{10}4+{\log }_{10}9=2{\log }_{10}\frac{a}{5}⟹{\log }_{10}(4\times 9)={\log }_{10}{a}^2-{\log }_{10}{5}^2⟹\log 36+\log 25=\log a^2⟹\log (36\times 25)=\log a^2\text{Removing log}{a}^2=36\times 25⟹a=6\times 5=30$.