Solve the logarithmic equation algebraically. Approximate the result to three decimal places. (If there is no solution, enter NO SOLUTION.) ln(x + 1) − ln(x − 4) = ln(x)

To solve the equation \( \ln(x + 1) - \ln(x - 4) = \ln(x) \) algebraically, we’ll start by applying logarithmic properties.

### Step 1: Use the Quotient Rule
The equation can be rewritten using the quotient rule for logarithms:
\[
\ln\left(\frac{x + 1}{x - 4}\right) = \ln(x)
\]

### Step 2: Eliminate the Logarithms
Since we have \( \ln\left(\frac{x + 1}{x - 4}\right) = \ln(x) \), we can exponentiate both sides to remove the logarithms:
\[
\frac{x + 1}{x - 4} = x
\]

### Step 3: Multiply Both Sides by \( x - 4 \)
To clear the fraction, multiply both sides by \( x - 4 \):
\[
x + 1 = x(x - 4)
\]
Expanding the right side:
\[
x + 1 = x^2 - 4x
\]

### Step 4: Rearrange into a Quadratic Equation
Bring all terms to one side of the equation to set it equal to zero:
\[
0 = x^2 - 5x - 1
\]

### Step 5: Solve the Quadratic Equation
The equation \( x^2 - 5x - 1 = 0 \) is a quadratic equation that can be solved using the quadratic formula:
\[
x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1}
\]
\[
x = \frac{5 \pm \sqrt{25 + 4}}{2}
\]
\[
x = \frac{5 \pm \sqrt{29}}{2}
\]

### Step 6: Simplify and Approximate
Calculating the two possible values for \( x \):
\[
x = \frac{5 + \sqrt{29}}{2} \approx 5.192
\]
\[
x = \frac{5 - \sqrt{29}}{2} \approx -0.192
\]

### Step 7: Check for Extraneous Solutions
Since we are dealing with logarithms, the arguments must be positive. Therefore, we need \( x > 4 \) for \( \ln(x - 4) \) to be defined.

- The solution \( x \approx 5.192 \) is valid since it is greater than 4.
- The solution \( x \approx -0.192 \) is not valid because it does not satisfy the domain restrictions.

### Final Answer
The solution is:
\[
x \approx 5.192
\]

Let me know if you need further details or have any questions!

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1. What are the steps to solving logarithmic equations involving the quotient rule?
2. How can extraneous solutions occur in logarithmic equations?
3. Why is the quadratic formula useful in solving quadratic equations?
4. How can logarithmic properties simplify equations?
5. What domain restrictions are commonly applied in logarithmic functions?

**Tip:** Always check for domain restrictions when solving logarithmic equations to avoid extraneous solutions.

Learn how to solve the logarithmic equation ln(x + 1) − ln(x − 4) = ln(x) using algebraic techniques. This problem covers logarithmic properties, the quotient rule, and solving quadratic equations. Approximate solutions to three decimal places. Suitable for students in Grades 9-12.

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