Question

Value of $x$ which satisfies the following relation $\frac{6}{5} a^{\log _a x \cdot \log _{10} a \cdot \log _a 5}=3^{\log _{10} \frac{x}{10}}+9^{\log _{100} x+\log _4 2}$. This value of $x$ lies between

• A. $(10,20)$
• B. $(30,40)$
• C. $(75,85)$
• D. $(95,105)$

Answer 1

Answer: (D)

$\frac{6}{5} x^{\log _{10} 5}=3^{\log _{10} x-1}+3^{\log _{10} x+1}$
$\frac{6}{5} 5^{\log _{10} x}=\frac{3^{\log _{10} x}}{3}+3 \cdot 3^{\log _{10} x}$
$\frac{6}{5} \cdot 5^t=\frac{3^t}{3}+3 \cdot 3^t $
$2 \cdot 3 \cdot 5^{t-1}=3^t \cdot\left(\frac{10}{3}\right) $
$5^{t-2}=3^{t-2}$
$\left(\frac{5}{3}\right)^{t-2}=\left(\frac{5}{3}\right)^0 $
$t=2 $
$\log _{10} x=2 $
$x=100$