Question
Solve for $\mathrm{x}$ and $\mathrm{y}$; if $\mathrm{x}>0$ and $\mathrm{y}>0 ; \log \mathrm{xy}=\log \frac{x}{y}+2 \log 2=2$
✓
Answer
$ \log x y=\log \left(\frac{x}{y}\right)+2 \log 2=2$
$ \log x y=2$
$ \Rightarrow \log x y=2 \log 10$
$ \Rightarrow \log x y=\log 10^2$
$ \Rightarrow \log x y=\log 100$
$\therefore x y=100\ldots(1)$
Now consider the equation
$\log \left(\frac{x}{y}\right)+2 \log 2=2$
$ \Rightarrow \log \left(\frac{x}{y}\right)+\log 2^2=2 \log 10$
$ \Rightarrow \log \left(\frac{x}{y}\right)+\log 4=\log 10^2$
$ \Rightarrow \log \left(\frac{x}{y}\right)+\log 4=\log 100$
$ \Rightarrow\left(\frac{x}{y}\right) \times 4=100$
$ \Rightarrow 4 x=100 y$
$ \Rightarrow x=25 y$
$ \Rightarrow x y=25 y x y$
$ \Rightarrow x y=25 y^2$
$\Rightarrow 100=25 y^2\dots ...[$ from $(1)]$
$ \Rightarrow \mathrm{y}^2=\frac{100}{25}$
$ \Rightarrow \mathrm{y}^2=4$
$\Rightarrow y=2 \ldots .[\because y>0]$
From $(1),$
$ x y=100$
$ \Rightarrow x \times 2=100$
$ \Rightarrow x=\frac{100}{2}$
$ \Rightarrow x=50 .$
Thus the values of $x$ and $y$ are $x=50$ and $y=2$.
$ \log x y=2$
$ \Rightarrow \log x y=2 \log 10$
$ \Rightarrow \log x y=\log 10^2$
$ \Rightarrow \log x y=\log 100$
$\therefore x y=100\ldots(1)$
Now consider the equation
$\log \left(\frac{x}{y}\right)+2 \log 2=2$
$ \Rightarrow \log \left(\frac{x}{y}\right)+\log 2^2=2 \log 10$
$ \Rightarrow \log \left(\frac{x}{y}\right)+\log 4=\log 10^2$
$ \Rightarrow \log \left(\frac{x}{y}\right)+\log 4=\log 100$
$ \Rightarrow\left(\frac{x}{y}\right) \times 4=100$
$ \Rightarrow 4 x=100 y$
$ \Rightarrow x=25 y$
$ \Rightarrow x y=25 y x y$
$ \Rightarrow x y=25 y^2$
$\Rightarrow 100=25 y^2\dots ...[$ from $(1)]$
$ \Rightarrow \mathrm{y}^2=\frac{100}{25}$
$ \Rightarrow \mathrm{y}^2=4$
$\Rightarrow y=2 \ldots .[\because y>0]$
From $(1),$
$ x y=100$
$ \Rightarrow x \times 2=100$
$ \Rightarrow x=\frac{100}{2}$
$ \Rightarrow x=50 .$
Thus the values of $x$ and $y$ are $x=50$ and $y=2$.
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