https://file.aiquickdraw.com/mathbot/file/1728627935843ywl8i98h.jpg

The equation presented in the image is:

\[
\log(x) + \log(x + 4) = 7
\]

Let's solve it step by step.

### Step 1: Use the logarithmic property
We can use the property of logarithms that states:

\[
\log(a) + \log(b) = \log(a \cdot b)
\]

This simplifies the left-hand side:

\[
\log(x) + \log(x + 4) = \log(x(x + 4)) = \log(x^2 + 4x)
\]

So, the equation becomes:

\[
\log(x^2 + 4x) = 7
\]

### Step 2: Remove the logarithm
To remove the logarithm, rewrite the equation in exponential form:

\[
x^2 + 4x = 10^7
\]

Simplifying \( 10^7 \):

\[
x^2 + 4x = 10,000,000
\]

### Step 3: Solve the quadratic equation
Rearrange the equation:

\[
x^2 + 4x - 10,000,000 = 0
\]

Now, we will solve this quadratic equation using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For this equation:

- \(a = 1\)
- \(b = 4\)
- \(c = -10,000,000\)

### Step 4: Apply the quadratic formula
Substituting into the quadratic formula:

\[
x = \frac{-4 \pm \sqrt{4^2 - 4(1)(-10,000,000)}}{2(1)}
\]

Simplifying the discriminant:

\[
x = \frac{-4 \pm \sqrt{16 + 40,000,000}}{2}
\]
\[
x = \frac{-4 \pm \sqrt{40,000,016}}{2}
\]
\[
x = \frac{-4 \pm 6324.5553}{2}
\]

### Step 5: Find the two solutions
Now we calculate the two possible values for \(x\):

1. \(x_1 = \frac{-4 + 6324.5553}{2} = 3160.2776\)
2. \(x_2 = \frac{-4 - 6324.5553}{2} = -3164.2776\)

Since logarithms are only defined for positive numbers, we discard the negative solution.

Thus, the valid solution is:

\[
x = 3160.2776
\]

Would you like more details or have any questions?

### Related Questions:
1. What are other properties of logarithms that can simplify equations?
2. Can you solve logarithmic equations using different methods?
3. What are the conditions where logarithmic equations have no solutions?
4. How do you solve logarithmic equations with different bases?
5. What is the general form of the quadratic formula?

### Tip:
Always check if a solution falls within the domain of the logarithmic function (i.e., the argument of the logarithm must be positive).

Learn how to solve the logarithmic equation log(x) + log(x + 4) = 7 with a detailed step-by-step solution. This problem explores logarithmic properties and solving quadratic equations using the quadratic formula. Suitable for students in Grades 10-12.

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