Given:
The given equation is [tex]x-4=2^{3}[/tex]
Solving the equation [tex]x-4=2^{3}[/tex], we get;
[tex]x-4=8[/tex]
[tex]x=12[/tex]
We need to determine the logarithmic equation that is equivalent to the given equation.
Option A: [tex]\log 3^{2}=(x-4)[/tex]
Simplifying, we get;
[tex]\log 9=x-4[/tex]
[tex]\log 9+4=x[/tex]
[tex]4.95=x[/tex]
Since, the values of x are not equivalent, the equation [tex]\log 3^{2}=(x-4)[/tex] is not equivalent to [tex]x-4=2^{3}[/tex]
Option A is not the correct answer.
Option B: [tex]\log 2^{3}=x-4[/tex]
Simplifying, we get;
[tex]\log 8=x-4[/tex]
[tex]\log 8+4=x[/tex]
[tex]4.9=x[/tex]
Since, the values of x are not equivalent, the equation [tex]\log 2^{3}=x-4[/tex] is not equivalent to [tex]x-4=2^{3}[/tex]
Option B is not the correct answer.
Option C: [tex]\log _{2}(x-4)=3[/tex]
Simplifying, we get;
[tex]x-4=2^{3}[/tex]
[tex]x-4=8[/tex]
[tex]x=12[/tex]
Since, the values of x are equivalent, the equation [tex]\log _{2}(x-4)=3[/tex] is equivalent to [tex]x-4=2^{3}[/tex]
Hence, Option C is the correct answer.
Option D: [tex]\log _{3}(x-4)=2[/tex]
Simplifying, we get;
[tex]x-4=3^2[/tex]
[tex]x-4=9[/tex]
[tex]x=13[/tex]
Since, the values of x are not equivalent, the equation [tex]\log _{3}(x-4)=2[/tex] is not equivalent to [tex]x-4=2^{3}[/tex]
Hence, Option D is not the correct answer.
