The solution for the logarithmic equation is x = 4 and it is graphically shown in the figure.
Every logarithmic function is the inverse function of an exponential function, we can think of every output on a logarithmic graph as the input for the corresponding inverse exponential equation.
For the given situation,
The equation is
[tex]log_{2}x+log_{2}(x-2) =3[/tex]
⇒ [tex]log_{2} (x)(x-2)=3[/tex]
⇒ [tex]log_{2}(x^{2} -2x)=3[/tex]
we know that [tex]y=log_{b} x[/tex] can be written as [tex]x=b^{y}[/tex]
⇒ [tex]x^{2} -2x=2^{3}[/tex]
⇒ [tex]x^{2} -2x-8=0[/tex]
⇒ [tex](x-4)(x+2)[/tex]
⇒ [tex]x=4[/tex] or [tex]x=-2[/tex]
The solution for the logarithmic function is shown in the graph below.
Hence we can conclude that the solution for the logarithmic function is x = 4.
Learn more about graphing a logarithmic function here
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