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Solve the following logarithmic equation. Express all solutions in exact form.√log x-3 =log x-3

Solve The Following Logarithmic Equation Express All Solutions In Exact Formlog X3 Log X3 class=

Sagot :

Square both side of equation and simplify the equation.

[tex]\begin{gathered} (\sqrt[]{\log x-3})^2=(\log x-3)^2 \\ \log x-3=(\log x)^2-6\log x+9 \\ (\log x)^2-6\log x-\log x+9+3=0 \\ (\log x)^2-7\log x+12=0 \end{gathered}[/tex]

Assume log x = y. So equation is,

[tex]y^2-7y+12=0[/tex]

Simplify the equation to obtain the value of y.

[tex]\begin{gathered} y^2-7y+12=0 \\ y^2-4y-3y+12=0 \\ y(y-4)-3(y-4)=0 \\ (y-3)(y-4)=0 \\ y=3,4 \end{gathered}[/tex]

So the value of y is 3 or 4,

[tex]\begin{gathered} \log x=3 \\ x=e^3 \end{gathered}[/tex]

Or

[tex]\begin{gathered} \log x=4 \\ x=e^4 \end{gathered}[/tex]

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