Order of steps to solve the given equation :
[tex]3^{log_3(x+2)}=3^{log_3(2x^2-1)}\\x+2=2x^2-1\\0=2x^2-x-3\\0=(2x-3)(x+1)\\x=-1 \; and \; \frac{3}{2}[/tex]
Given :
[tex]log_3(x+2)=log_3(2x^2-1)[/tex]
In step 1, We take base 3 on both sides of the given equation
[tex]3^{log_3(x+2)}=3^{log_3(2x^2-1)}[/tex]
Both base 3 and log have base 3. So we equate the log terms and then we solve for x
[tex]x+2=2x^2-1[/tex]
In the next step , move all terms to the right side and make the equation =0
[tex]x+2=2x^2-1\\x+2-x-2=2x^2-1-x-2\\0=2x^2-x-3[/tex]
Now , we factor it and solve for x
[tex]0=2x^2-x-3\\0=(2x-3)(x+1)[/tex]
Set each factor =0 and solve for x
[tex]0=(2x-3)(x+1)\\2x-3=0, x=\frac{3}{2} \\\\x+1=0, x=-1[/tex]
Both values of x satisfies the given equation .
Both the x values are the solutions
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