Jerome solved the equation below by graphing.log2(x) + log2(x-2) = 3Which of the following shows the correct system of equations and solution?

Accepted Solution

Answer:B. x = 4Step-by-step explanation:I can't speak to the first part of this question, as I don't totally have context for what they're asking, but we can solve for x using one of the laws of logarithms, namely:[tex]\log_bm+\log_bn=\log_bmn[/tex]Using this law, we can combine and rewrite our initial equation as[tex]\log_2(x\cdot(x-2))=3\\\log_2(x^2-2x)=3[/tex]Remember that logarithms are simply another way of writing exponents. The logarithm [tex]\log_28=3[/tex] is just another way of writing the fact [tex]2^3=8[/tex]. Keeping that in mind, we can express our logarithm in terms of exponents as[tex]\log_2(x^2-2x)=3\rightarrow2^3=x^2-2x[/tex]2³ = 8, so we can replace the left side of our equation with 8 to get[tex]8 = x^2-2x[/tex]Moving the 8 to the other side:[tex]0=x^2-2x-8[/tex]We can now factor the expression on the right to find solutions for x:[tex]0=(x-4)(x+2)\\x=4, -2[/tex]The only option which agrees with our solution is B.