https://file.aiquickdraw.com/mathbot/file/1729014594968azbbgc04.jpg

The equation in the image is:

\[
\log_2(x + 6) - \log_2(x + 2) = \log_2(3)
\]

### Step 1: Apply logarithmic properties
We can use the **logarithmic subtraction rule**, which states:
\[
\log_b(A) - \log_b(B) = \log_b\left(\frac{A}{B}\right)
\]
Thus, the left side becomes:
\[
\log_2\left(\frac{x+6}{x+2}\right)
\]

Now the equation looks like this:
\[
\log_2\left(\frac{x+6}{x+2}\right) = \log_2(3)
\]

### Step 2: Rewrite in exponential form
Since the logarithms have the same base (base 2), we can set the arguments equal to each other:
\[
\frac{x+6}{x+2} = 3
\]

### Step 3: Solve the equation
To solve for \(x\), first multiply both sides by \(x + 2\) to eliminate the denominator:
\[
x + 6 = 3(x + 2)
\]

Distribute the 3 on the right-hand side:
\[
x + 6 = 3x + 6
\]

Now, subtract \(x\) and 6 from both sides:
\[
0 = 2x
\]

Thus, the solution is:
\[
x = 0
\]

### Conclusion:
The correct exponential form of the equation is:
\[
\frac{x+6}{x+2} = 3
\]
And the solution is \(x = 0\).

Would you like more details on any step or have any questions?

Here are 5 related questions to consider:
1. What happens if we change the base of the logarithms in the equation?
2. Can you explain the other logarithmic properties used to manipulate logarithmic expressions?
3. How does the solution change if the constant on the right-hand side is different from 3?
4. How can logarithmic equations be solved graphically?
5. What are some real-life applications of solving logarithmic equations?

**Tip**: Always remember to check the domain of the logarithmic function to ensure the arguments inside the logs are positive.

To solve the equation log_2(x + 6) - log_2(x + 2) = log_2(3) one must use properties of the logarithms and rewrite the equation in exponential form. What is the correct form?

Learn how to solve the logarithmic equation log_2(x + 6) - log_2(x + 2) = log_2(3) by applying logarithmic properties and rewriting it in exponential form. This step-by-step solution covers important concepts of logarithms and exponents, suitable for Grades 9-12.

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