Explain why or why not. Determine whether the following statements are true and give an explanation or counterexample.


b. ln(x + 1) + ln(x − 1) = ln(x² − 1), for all x.

Verified step by step guidance

1. Step 1: Recall the logarithmic property that states \( \ln(a) + \ln(b) = \ln(ab) \). This property allows us to combine the logarithms on the left-hand side of the equation.
2. Step 2: Apply the property from Step 1 to the left-hand side: \( \ln(x + 1) + \ln(x - 1) = \ln((x + 1)(x - 1)) \).
3. Step 3: Simplify the expression \((x + 1)(x - 1)\) using the difference of squares formula: \((x + 1)(x - 1) = x^2 - 1\).
4. Step 4: Substitute the simplified expression from Step 3 back into the equation: \( \ln((x + 1)(x - 1)) = \ln(x^2 - 1) \).
5. Step 5: Conclude that the original statement \( \ln(x + 1) + \ln(x - 1) = \ln(x^2 - 1) \) is true for all \( x \) such that \( x > 1 \) or \( x < -1 \), because the domain of the logarithmic function requires that the arguments \( x + 1 \) and \( x - 1 \) are positive.

Key Concepts

Properties of Logarithms

Logarithms have specific properties that govern their behavior, such as the product, quotient, and power rules. The product rule states that ln(a) + ln(b) = ln(ab), while the quotient rule states that ln(a) - ln(b) = ln(a/b). Understanding these properties is essential for manipulating logarithmic expressions and verifying the validity of statements involving logarithms.

Domain of Logarithmic Functions

The domain of a logarithmic function is restricted to positive real numbers. For the expression ln(x + 1) + ln(x - 1), both x + 1 and x - 1 must be greater than zero, which implies x must be greater than 1. Recognizing the domain is crucial for determining the validity of logarithmic equations and ensuring that all terms are defined.

Counterexamples in Mathematics

A counterexample is a specific case that disproves a general statement. In the context of the given logarithmic equation, finding a value of x that makes the left-hand side unequal to the right-hand side serves as a counterexample. This method is vital in mathematical reasoning, as it helps to establish the truth or falsehood of statements by demonstrating exceptions.