Question

Let $ a =\left(\log _3 \pi\right)\left(\log _2 3\right)\left(\log _\pi 2\right)$,
$b =\frac{\log 576}{3 \log 2+\log 3}$ the base of the logarithm being 10 , $c =2\left(\right.$ sum of the solution of the equation $\left.(3)^{4 x }-(3)^{\left(2 x +\log _3(12)\right)}+27=0\right)$ and $d =7^{\left(\log _7 2+\log _7 3\right)}$ then $(a+b+c \div d)$ simplifies to

• A. rational which is not natural.
• B. natural but not prime.
• C. irrational
• D. even but not composite

Answer 1

Answer: (A)

$a =\left(\log _3 \pi\right)\left(\log _2 3\right)\left(\log _\pi 2\right)=\frac{\log \pi}{\log 3} \cdot \frac{\log 3}{\log 2} \cdot \frac{\log 2}{\log \pi}=1 $
$b =\frac{\log 576}{3 \log 2+\log 3} \text { the base of the log being } 10=\frac{2 \log 24}{\log 24}=2 $
$c =(9)^{2 x }-9^{ x } \cdot 3^{\log _3 12}+27=0 \Rightarrow t ^2-12 t +27=0$
$\Rightarrow t =9 \Rightarrow 9^{ x }=9 \Rightarrow x =1$
$\Rightarrow t =3 \Rightarrow 9^{ x }=3 \Rightarrow x =\frac{1}{2} $
$\Rightarrow c =2\left(1+\frac{1}{2}\right)=3 $
$d =7^{\left(\log _7 2+\log _7 3\right)}=6 $
$a + b +\frac{ c }{ d }=1+2+\frac{3}{6}=\frac{7}{2} $