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Option 2 is correct. i.e. x = 1
What is the correct answer ?
The given logarithmic equation is
[tex]log_{2}(x)+ log_{2}(x+7)=3[/tex]
⇒ [tex]log_{2}[x(x+7]=3[/tex] [∵[tex]loga+logb=log ab[/tex]]
⇒ [tex]x(x+7)=2^{3}[/tex]
⇒ [tex]x^{2} +7x-8=0[/tex]
⇒ [tex](x+8)(x-1)=0[/tex]
⇒ x = -8, 1
At, x = 1
[tex]log_{2}(1)+ log_{2}(1+7)=3[/tex]
⇒ 0+3=3
⇒ 3=3
L.H.S = R.H.S, therefore x=1 is a solution of the given equation.
At, x = -8
[tex]log_{2}(-8)+ log_{2}(-8+7)=3[/tex]
⇒ [tex]log_{2}(-8)+ log_{2}(-1)=3[/tex]
Logarithmic functions are defined only for positive values.
So, x = -8 is not a solution of the given equation, i.e. a extraneous solution.
Hence, x= 1 is the only solution of the given equation.
Learn more about logarithmic function here :
https://brainly.com/question/27845085
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