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(5x) + ln 1 = ln(5x)

Key Concepts

Properties of Logarithms

Logarithms have specific properties that simplify expressions. One key property is that the logarithm of a product is the sum of the logarithms of the factors, expressed as ln(a) + ln(b) = ln(ab). Understanding these properties is essential for manipulating logarithmic equations and determining their validity.

Natural Logarithm (ln)

The natural logarithm, denoted as ln, is the logarithm to the base e, where e is approximately 2.718. It is commonly used in calculus and algebra due to its unique properties, such as the fact that the derivative of ln(x) is 1/x. Recognizing how ln operates is crucial for solving equations involving natural logarithms.

Identity in Logarithmic Equations

An identity in mathematics is an equation that holds true for all values of the variable involved. In logarithmic equations, verifying whether both sides of the equation are equal under certain conditions is vital. If an equation is not an identity, adjustments must be made to establish a true statement, which is the focus of the given exercise.