Question

If equations $\log _{128} x +\log _{125} y =5$ and $\log _{ y } 125-\log _{ x } 128=1$ is satisfied by ordered pairs $\left( x _1\right.$, $\left.y_1\right)$ and $\left(x_2, y_2\right)$, then find the value of $\log _{21}\left(\log _{10} x_1 x_2 y_1 y_2\right)$.

Answer 1

Let $\log _{128} x$ and $\log _{125} y = b$
$\therefore a+b=5 \Rightarrow b=5-a $
$\text { and } \frac{1}{b}-\frac{1}{a}=1 \Rightarrow \frac{a-b}{a b}=1 \Rightarrow a-b=a b$
$\Rightarrow a-(5-a)=a(5-a) \Rightarrow a^2-3 a-5=0 $
$\therefore a_1+a_2=3$
$\Rightarrow \log _{128} x_1+\log _{128} x_2=3 \Rightarrow x_1 x_2=(128)^3=2^{21} $
$\Theta b=5-a$
$\therefore b_1+b_2=10-\left(a_1+a_2\right)=7 $
$\Rightarrow \log _{125}\left(y_1 y_2\right)=7 \Rightarrow y_1 y_2=5^{21} $
$\therefore x_1 x_2 y_1 y_2=10^{21} $
$\Rightarrow \log _{10}\left(x_2 y_1 y_2\right)=21 $
$\Rightarrow \log _{21}\left(\log _{10} x_1 x_2 y_1 y_2\right)=1 $