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Order the steps to solve the equation [tex]\log (x^2 - 15) = \log (2x)[/tex] from 1 to 5.

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]


Sagot :

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].