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[Solved] Logarithm: Solve the system of equations

 
Logorithm
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June 15, 2021 2:44 pm
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(@armor)
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Solve the system of equations

lg |x+y|=1,

lgy-lg|x|=\frac{1}{log_{4}100}.

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June 15, 2021 2:44 pm
Topic starter
Staff 
(@armor)
Admin
  Honorable Member
317 Posts
156 153 0

The system is equivalent to 

\small lg|x+y|=lg10

\small lg\frac{y}{|x|}=lg2

which leads to  

\small |x+y|=10,

\small y=2|x|.

\small y>0  is always true since \small y=2|x|  and  \small x\neq 0.

When \small x>0,  the system becomes 

\small x+y=10,

\small y=2x,

whose solution is x=10/3, y=20/3.

When \small x<0,  the system becomes

\small x+y=10,

\small y=-2x,

whose solution is x=-10, y=20.

We can verify that (10/3, 20/3), (-10,20) indeed are solutions of the original system.

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