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).