To solve the equation:

$\log_2(x+7) - \log_2(x+1) = 3$

Step 1: Apply the logarithmic property

Using the property $\log_b(A) - \log_b(B) = \log_b\left(\frac{A}{B}\right)$, the equation becomes:

$\log_2\left(\frac{x+7}{x+1}\right) = 3$

Step 2: Rewrite in exponential form

Rewrite the logarithmic equation in its exponential form:

$\frac{x+7}{x+1} = 2^3$

Since $2^3 = 8$, the equation becomes:

$\frac{x+7}{x+1} = 8$

Step 3: Eliminate the fraction

Multiply both sides of the equation by $x+1$ (assuming $x+1 \neq 0$):

$x+7 = 8(x+1)$

Step 4: Expand and simplify

Expand the right-hand side:

$x+7 = 8x + 8$

Simplify:

$7 - 8 = 8x - x$

$-1 = 7x$

Step 5: Solve for $x$

Divide both sides by 7:

$x = -\frac{1}{7}$

Final Answer:

$x = -\frac{1}{7}$

Let me know if you have any questions or want a step-by-step clarification.

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Follow-up questions:

1. How do logarithmic properties help simplify equations?
2. What is the significance of converting logarithmic equations to exponential form?
3. What are common errors to avoid when solving logarithmic equations?
4. Can logarithms with different bases be combined in similar ways?
5. How can you verify solutions to logarithmic equations?

Tip: Always check if the solution lies within the domain of the logarithmic function to avoid undefined values!