To order the steps to solve the equation [tex]\(\log \left(x^2-15\right)=\log(2x)\)[/tex] from 1 to 5, follow these steps:
1. Start with the given equation [tex]\(\log \left(x^2-15\right) = \log (2 x)\)[/tex].
Since [tex]\(\log(a) = \log(b)\)[/tex] implies [tex]\(a = b\)[/tex], we can set the arguments of the logarithms equal to each other:
[tex]\[
x^2 - 15 = 2x
\][/tex]
2. Rearrange the equation to standard quadratic form:
[tex]\[
x^2 - 2x - 15 = 0
\][/tex]
3. Factorize the quadratic equation:
[tex]\[
(x - 5)(x + 3) = 0
\][/tex]
4. Solve for [tex]\(x\)[/tex] by setting each factor equal to zero:
[tex]\[
x - 5 = 0 \text { or } x + 3 = 0
\][/tex]
5. Find the potential solutions:
[tex]\[
x = 5 \text { or } x = -3
\][/tex]
Thus, the ordered steps are:
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]\(x - 5 = 0 \text { or } x + 3 = 0\)[/tex]
5. Potential solutions are -3 and 5
