The given equation can be written in the form:Nowis defined for 0 src=https://files.askiitians.com/cdn1/cms-content/common/latex.codecogs.comgif.latexxgt0.jpg>. So, 0 \\quad and \\quad \\frac{3-x}{x-1} > 0 src=https://files.askiitians.com/cdn1/cms-content/common/latex.codecogs.comgif.latex_fracx4xgt0_quadand_quad_frac3-xx-1gt0.jpg>Solving these two we get…......................(1)To solve, assume left hand side as and right hand side as. Then we get.….................(2)From (1) and (2)is a possible solution.Similarly solving, we get..........(3)From (1) and (3)is a possible solution.Hence we get two solutions to the given equation:.

Grade 12Algebra

question is in image

Aman Gandhi

11 Years agoGrade 12

Satyajit Samal

11 Years ago

The given equation can be written in the form:
$\log ^{ 2 }{ \left( \frac { x+4 }{ x } \right) }+\log ^{ 2 }{ \left( \frac { x }{ x+4 } \right) }=2\log ^{ 2 }\left( { \frac { 3-x }{ x-1 } } \right) \\ \Rightarrow 2\log ^{ 2 }{ \left( \frac { x+4 }{ x } \right) }=2\log ^{ 2 }\left( { \frac { 3-x }{ x-1 } } \right)$
Now $\log x$ is defined for $x > 0$ . So, $\frac{x+4}{x}> 0 \quad and \quad \frac{3-x}{x-1} > 0$
Solving these two we get $1 < x < 3$ …......................(1)
To solve $\log ^{ 2 }{ \left( \frac { x+4 }{ x } \right) }=\log ^{ 2 }\left( { \frac { 3-x }{ x-1 } } \right)$ , assume left hand side as $t^{2}$ and right hand side as $m^{2}$ . Then we get $\left ( t-m \right )\left ( t+m \right )=0 \\ \Rightarrow t=m\; or\; t=-m$ .
$\log { \left( \frac { x+4 }{ x } \right) }=\log \left( { \frac { 3-x }{ x-1 } } \right) \\ \Rightarrow \log \left ( \frac{x+4}{x}.\frac{x-1}{3-x} \right )=0 \\ \Rightarrow x = \pm \sqrt{2}$ ….................(2)
From (1) and (2) $x= \sqrt{2}$ is a possible solution.
Similarly solving $\log { \left( \frac { x+4 }{ x } \right) }=-\log \left( { \frac { 3-x }{ x-1 } } \right)$ , we get $x= \pm \sqrt {6}$ ..........(3)
From (1) and (3) $x= \sqrt {6}$ is a possible solution.
Hence we get two solutions to the given equation: $\sqrt{2}\;, \sqrt {6}$ .

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Aman Gandhi

1 Answer

Satyajit Samal

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question is in image

Aman Gandhi

1 Answer

Satyajit Samal

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