Find the value of x: ln(x+1) - ln(x-5) = ln3

To solve the equation \( \ln(x+1) - \ln(x-5) = \ln 3 \), follow these steps:

### Step 1: Apply the logarithm property
Use the property of logarithms: \( \ln a - \ln b = \ln\left(\frac{a}{b}\right) \). Applying this to the equation gives:
\[
\ln\left(\frac{x+1}{x-5}\right) = \ln 3
\]

### Step 2: Exponentiate both sides
To eliminate the logarithms, exponentiate both sides with base \( e \):
\[
\frac{x+1}{x-5} = 3
\]

### Step 3: Solve the resulting equation
Now, solve the equation \( \frac{x+1}{x-5} = 3 \) for \( x \).

Multiply both sides by \( x - 5 \) to get rid of the fraction:
\[
x + 1 = 3(x - 5)
\]

Distribute the \( 3 \) on the right side:
\[
x + 1 = 3x - 15
\]

### Step 4: Isolate \( x \)
Move \( x \) to one side of the equation by subtracting \( x \) from both sides:
\[
1 + 15 = 3x - x
\]

Simplify:
\[
16 = 2x
\]

Finally, divide by 2 to solve for \( x \):
\[
x = 8
\]

### Final Answer
The value of \( x \) is \( \boxed{8} \).

Would you like more details or have any questions? Here are some related questions:

1. What are the logarithm properties used in solving this problem?
2. How do you solve logarithmic equations involving multiple logarithms?
3. Can you solve for \( x \) if the equation was \( \ln(x+2) - \ln(x-4) = \ln 2 \)?
4. What happens if the logarithmic equation has no solution?
5. How can you verify the solution of a logarithmic equation?
6. What are the domain restrictions for logarithmic equations?
7. How do logarithmic functions behave compared to exponential functions?
8. How do you differentiate a logarithmic function?

**Tip:** Always check the domain of the original logarithmic functions to ensure the solution is valid (i.e., arguments of logarithms must be positive).

Learn how to solve the logarithmic equation ln(x+1) - ln(x-5) = ln 3 step-by-step. This problem involves applying logarithm properties and solving exponential equations. Suitable for high school students.

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