In Exercises 60–63, 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)(ln 1) = 0

Key Concepts

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 algebra and calculus, often used to solve equations involving exponential growth or decay. Understanding the properties of logarithms, such as ln(1) = 0, is crucial for evaluating expressions involving ln.

Properties of Logarithms

Logarithmic properties are rules that simplify the manipulation of logarithmic expressions. Key properties include the product rule, quotient rule, and power rule. For instance, the property ln(a) + ln(b) = ln(ab) helps in combining logarithmic terms, while ln(1) = 0 is essential for evaluating expressions where the logarithm of one is involved.

Equation Evaluation

Evaluating equations involves determining the truth value of a mathematical statement. In this context, it requires substituting known values and applying algebraic rules to verify if both sides of the equation are equal. If the equation is found to be false, one must identify the necessary adjustments to create a true statement, which is a critical skill in algebra.