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

Key Concepts

Properties of Logarithms

Logarithms have specific properties that govern their behavior, including 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 determining the validity of equations involving logarithms.

Natural Logarithm (ln)

The natural logarithm, denoted as ln, is the logarithm to the base e, where e is approximately 2.71828. It is a fundamental concept in calculus and algebra, particularly in solving exponential equations. Recognizing that ln(1) equals 0 is crucial for simplifying expressions and evaluating the truth of logarithmic equations.

Domain of Logarithmic Functions

Logarithmic functions are defined only for positive arguments. This means that for ln(x + 1) and ln(x), the expressions inside the logarithm must be greater than zero. Understanding the domain restrictions is vital for determining the validity of logarithmic equations and ensuring that all operations performed are mathematically sound.