The equation given in the image is:

$\ln(x + 1) - \ln(x - 3) = \ln 4$

Step-by-step solution:

1. Apply the logarithmic property: $\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)$ Using this property, the equation becomes: $\ln\left(\frac{x + 1}{x - 3}\right) = \ln 4$

2. Equate the arguments of the logarithms: If $\ln A = \ln B$, then $A = B$. Therefore: $\frac{x + 1}{x - 3} = 4$

3. Simplify the equation: Multiply both sides by $x - 3$ (assuming $x - 3 \neq 0$): $x + 1 = 4(x - 3)$

   Expand the right-hand side: $x + 1 = 4x - 12$

4. Solve for $x$: Rearrange to isolate $x$: $1 + 12 = 4x - x$ $13 = 3x$ $x = \frac{13}{3}$

5. Check the solution: Ensure $x = \frac{13}{3}$ satisfies the domain of the logarithmic functions:

   • The arguments of the logarithms must be positive: $x + 1 > 0 \quad \text{and} \quad x - 3 > 0$ For $x + 1 > 0$, $x > -1$.
     For $x - 3 > 0$, $x > 3$.
     The stricter condition is $x > 3$.

   Since $\frac{13}{3} \approx 4.33 > 3$, the solution is valid.

Final Answer:

$x = \frac{13}{3}$

Would you like further clarification or details on this solution?

Related Questions:

1. What is the domain of the given logarithmic equation?
2. How can the logarithmic property $\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)$ be derived?
3. How would the solution change if the right-hand side was a different value (e.g., $\ln 2$)?
4. Why is it important to check the domain of logarithmic equations?
5. Can this equation be solved graphically? How?

Tip: Always check the domain restrictions when working with logarithmic functions to avoid invalid solutions.