Solve the equation step by step    ln(x+1)+ln(x)=ln(2)

We are tasked with solving the equation:

\[
\ln(x + 1) + \ln(x) = \ln(2)
\]

### Step 1: Use the logarithmic property
We can use the logarithmic property:

\[
\ln(a) + \ln(b) = \ln(a \cdot b)
\]

Thus, the left-hand side of the equation becomes:

\[
\ln((x + 1) \cdot x) = \ln(2)
\]

This simplifies to:

\[
\ln(x(x + 1)) = \ln(2)
\]

### Step 2: Remove the natural logarithms
Since the natural logarithm functions on both sides of the equation are equal, the arguments must be equal. Therefore:

\[
x(x + 1) = 2
\]

### Step 3: Expand the equation
Now, expand the left-hand side:

\[
x^2 + x = 2
\]

### Step 4: Rearrange the equation into a quadratic form
Move all terms to one side to form a standard quadratic equation:

\[
x^2 + x - 2 = 0
\]

### Step 5: Solve the quadratic equation
We can solve this quadratic equation using factoring. We look for two numbers that multiply to \(-2\) and add to \(1\). These numbers are \(2\) and \(-1\), so we factor the quadratic as:

\[
(x - 1)(x + 2) = 0
\]

### Step 6: Find the solutions
Set each factor equal to zero:

\[
x - 1 = 0 \quad \text{or} \quad x + 2 = 0
\]

This gives us the solutions:

\[
x = 1 \quad \text{or} \quad x = -2
\]

### Step 7: Check for extraneous solutions
Since the logarithmic function is only defined for positive arguments, we must check the domain of the original equation. Both \(\ln(x + 1)\) and \(\ln(x)\) require \(x > 0\). Therefore, the solution \(x = -2\) is not valid.

Thus, the only valid solution is:

\[
x = 1
\]

Would you like further clarification on any step?

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Here are 5 related questions to deepen your understanding:

1. How do we solve logarithmic equations with different bases?
2. What is the domain of a logarithmic function?
3. Can logarithmic equations have no solutions? If so, under what conditions?
4. How can we solve a quadratic equation that cannot be factored easily?
5. What are the key properties of logarithms that help in solving equations?

**Tip:** Always check the domain of logarithmic functions to ensure the solution is valid!

Solve the equation step by step ln(x+1) + ln(x) = ln(2)

Learn how to solve the logarithmic equation ln(x+1) + ln(x) = ln(2) with a step-by-step guide. This solution covers logarithmic properties and quadratic equation concepts. Suitable for students in Grades 9-11.

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