In Exercises 89–102, determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. ln(x + 1) = ln x + ln 1

Verified step by step guidance

1. Recall the logarithm property: \( \ln(ab) = \ln a + \ln b \). This means the sum of logarithms corresponds to the logarithm of a product.
2. Compare the given equation \( \ln(x + 1) = \ln x + \ln 1 \) to the property. The right side is \( \ln x + \ln 1 = \ln(x \cdot 1) = \ln x \).
3. Since \( \ln(x + 1) \) is on the left and \( \ln x \) is on the right, the equation simplifies to \( \ln(x + 1) = \ln x \).
4. For \( \ln(x + 1) = \ln x \) to be true, the arguments must be equal: \( x + 1 = x \), which is impossible for any real \( x \).
5. Therefore, the original equation is false. To make it true, replace \( \ln 1 \) with \( \ln(x + 1) - \ln x \), or rewrite the right side as \( \ln(x + 1) = \ln x + \ln\left(\frac{x + 1}{x}\right) \).

Key Concepts

Properties of Logarithms

Logarithms have specific properties that govern their operations. One key property is that the logarithm of a product equals the sum of the logarithms: ln(ab) = ln a + ln b. Understanding these properties helps determine if an equation involving logarithms is true or false.

Domain of Logarithmic Functions

The domain of a logarithmic function ln(x) includes only positive real numbers (x > 0). When evaluating or manipulating logarithmic expressions, it is essential to ensure that all arguments inside the logarithms are positive to avoid undefined expressions.

Equation Verification and Manipulation

To verify if an equation is true, substitute values or apply algebraic properties to simplify both sides. If false, identify the incorrect step and adjust the equation accordingly. This process is crucial for validating or correcting logarithmic equations.