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 + ln(2x) = ln(3x)

Verified step by step guidance

1. Step 1: Recall the property of logarithms that states \( \ln(a) + \ln(b) = \ln(ab) \).
2. Step 2: Apply this property to the left side of the equation: \( \ln(x) + \ln(2x) = \ln(x \cdot 2x) = \ln(2x^2) \).
3. Step 3: Compare the transformed left side \( \ln(2x^2) \) with the right side \( \ln(3x) \).
4. Step 4: Recognize that for the equation to be true, the expressions inside the logarithms must be equal: \( 2x^2 = 3x \).
5. Step 5: Solve the equation \( 2x^2 = 3x \) by factoring or using algebraic methods to determine if there are values of \( x \) that satisfy the equation.

Key Concepts

Properties of Logarithms

Logarithms have specific properties that simplify expressions. Key properties include the product rule, which states that ln(a) + ln(b) = ln(ab), and the quotient rule, ln(a) - ln(b) = ln(a/b). Understanding these properties is essential for manipulating logarithmic equations and verifying 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 exponential growth models. Recognizing how ln interacts with exponential functions is crucial for solving equations involving ln.

Equation Verification

To determine if an equation is true or false, one must verify both sides of the equation under the same conditions. This often involves substituting values or simplifying both sides to see if they are equal. If they are not, identifying necessary changes to make the equation true is a key skill in algebra.