Logaritmic equation

Logarithms problems.

Logaritmic equation

Postby Guest » Tue Sep 15, 2020 1:51 pm

Greetings.

I need your help solving this logaritmic equation.

I tried to equal the base of each log then convert it into one only log and i got to the result 2^(x^2-4)=x+3/x-3
Your help is deeply appreciated.
thank you.
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Re: Logaritmic equation

Postby Guest » Wed Sep 16, 2020 12:58 am

[tex]x_{1 }[/tex]=1 ,[tex]x_{2 }[/tex]=2

Is that the answer ?
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Re: Logaritmic equation

Postby Guest » Thu Sep 17, 2020 12:53 pm

[tex]x^{2}[/tex][tex]log_{2 }[/tex][tex]\frac{3+x}{10}[/tex]-[tex]x^{2}[/tex][tex]log_{\frac{1}{2} }[/tex](2+3x)=[tex]x^{2}[/tex]-4+2[tex]log_{\sqrt{2} }[/tex][tex]\frac{3x^{2}+11x+6}{10}[/tex]

Definition set
[tex]\begin{array}{|l} \frac{3+x}{10} >0 \\ 2+3x>0\\\frac{3x^{2}+11x+6}{10} >0\end{array}[/tex]

[tex]\begin{array}{|l} x >-3 \\ x >-\frac{2}{3}\\x\in(-\infty;-3)\cup (-\frac{2}{3};+\infty)\end{array}[/tex] [tex]\Rightarrow[/tex] x[tex]\in[/tex](-[tex]\frac{2}{3}[/tex];+[tex]\infty[/tex]) :!:
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[tex]x^{2}[/tex][tex]log_{2 }[/tex][tex]\frac{3+x}{10}[/tex]+[tex]x^{2}[/tex][tex]log_{2 }[/tex](2+3x)-2.2[tex]log_{2 }[/tex][tex]\frac{(3+x)(3x+2)}{10}[/tex]=[tex]x^{2}[/tex]-4

[tex]x^{2}[/tex][ [tex]log_{2 }[/tex][tex]\frac{3+x}{10}[/tex]+[tex]log_{2 }[/tex](2+3x) ]-4[tex]log_{2 }[/tex][tex]\frac{(3+x)(3x+2)}{10}[/tex]=[tex]x^{2}[/tex]-4

[tex]x^{2}[/tex][tex]log_{2 }[/tex][tex]\frac{(3+x)(3x+2)}{10}[/tex]-4[tex]log_{2 }[/tex][tex]\frac{(3+x)(3x+2)}{10}[/tex]=[tex]x^{2}[/tex]-4

([tex]x^{2}[/tex]-4)[tex]log_{2 }[/tex][tex]\frac{(3+x)(3x+2)}{10}[/tex]=[tex]x^{2}[/tex]-4

1 case [tex]x_{1 }[/tex]=-2 [tex]\notin[/tex] Def.set
2 case [tex]x_{2 }[/tex]=2 [tex]\in[/tex] Def.set
3 case x[tex]\ne[/tex][tex]\pm[/tex]2 [tex]\Rightarrow[/tex] [tex]log_{2 }[/tex][tex]\frac{(3+x)(3x+2)}{10}[/tex]=1

[tex]\frac{9x+6+3x^{2}+2x}{10}[/tex]=[tex]2^{1}[/tex] [tex]\Rightarrow[/tex] 3[tex]x^{2}[/tex]+11x-14=0

D=121-4.3(-14) =289=[tex]17^{2}[/tex] ; [tex]x_{3,4 }[/tex]=[tex]\frac{-11\pm17}{2.3}[/tex]

[tex]x_{3 }[/tex]=-[tex]\frac{28}{6}[/tex]= -4,(6) [tex]\notin[/tex]Def.set
[tex]x_{4 }[/tex]=[tex]\frac{6}{6}[/tex]=1 [tex]\in[/tex] Def.set
The answers [tex]x_{1 }[/tex] ,[tex]x_{3 }[/tex]-are dropped .

Answer: 1:2
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