Sure, let's order the steps to solve the equation [tex]\(\log (x^2 - 15) = \log (2x)\)[/tex].
1. Remove the logarithms by equating the arguments:
[tex]\[
x^2 - 15 = 2x
\][/tex]
2. Bring all terms to one side to form a quadratic equation:
[tex]\[
x^2 - 2x - 15 = 0
\][/tex]
3. Factorize the quadratic equation:
[tex]\[
(x - 5)(x + 3) = 0
\][/tex]
4. Solve for the potential solutions:
[tex]\[
x - 5 = 0 \quad \text{or} \quad x + 3 = 0 \\
x = 5 \quad \text{or} \quad x = -3
\][/tex]
5. Verify potential solutions satisfy the original equation:
[tex]\[
\text{Potential solutions are } -3 \text{ and } 5
\][/tex]
Now, putting these steps in the correct order:
1. [tex]\[
x^2 - 15 = 2x
\][/tex]
2. [tex]\[
x^2 - 2x - 15 = 0
\][/tex]
3. [tex]\[
(x - 5)(x + 3) = 0
\][/tex]
4. [tex]\[
x - 5 = 0 \quad \text{or} \quad x + 3 = 0 \\
(x-5)(x+3)=0
\][/tex]
5. [tex]\[
\text{Potential solutions are } -3 \text{ and } 5
\][/tex]
