Sure! Here is a detailed, step-by-step solution for solving the equation [tex]\(\log(x^2 - 15) = \log(2x)\)[/tex]:
### Step 1
Use the property of logarithms which states that if [tex]\(\log A = \log B\)[/tex], then [tex]\(A = B\)[/tex]. Applying this property to our equation:
[tex]\[
x^2 - 15 = 2x
\][/tex]
### Step 2
Rearrange the equation to form a standard quadratic equation by subtracting [tex]\(2x\)[/tex] from both sides:
[tex]\[
x^2 - 2x - 15 = 0
\][/tex]
### Step 3
Factor the quadratic equation. Factoring is the process of expressing the equation as a product of binomials:
[tex]\[
(x - 5)(x + 3) = 0
\][/tex]
### Step 4
Set each factor equal to zero and solve for [tex]\(x\)[/tex]. This step will give us the potential solutions:
[tex]\[
x - 5 = 0 \quad \text{or} \quad x + 3 = 0
\][/tex]
### Step 5
Solve the simple equations from the previous step to find the potential solutions for [tex]\(x\)[/tex]:
[tex]\[
x = 5 \quad \text{or} \quad x = -3
\][/tex]
Now, let's order the given steps from the problem according to the logical sequence we have established above:
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]\(x - 5 = 0\)[/tex] or [tex]\(x + 3 = 0\)[/tex]
5. Potential solutions are [tex]\(x = 5\)[/tex] and [tex]\(x = -3\)[/tex]
These steps will guide you through solving the equation [tex]\(\log(x^2 - 15) = \log(2x)\)[/tex].
