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(8x^3) = 3 ln (2x)

Verified step by step guidance

1. Step 1: Recall the logarithmic identity \( \ln(a^b) = b \ln(a) \).
2. Step 2: Apply the identity to the left side: \( \ln(8x^3) = \ln((2^3)(x^3)) = \ln(2^3) + \ln(x^3) = 3\ln(2) + 3\ln(x) \).
3. Step 3: Simplify the right side: \( 3 \ln(2x) = 3(\ln(2) + \ln(x)) = 3\ln(2) + 3\ln(x) \).
4. Step 4: Compare both sides: \( 3\ln(2) + 3\ln(x) = 3\ln(2) + 3\ln(x) \).
5. Step 5: Conclude that the equation is true, as both sides are equal.

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms is essential for manipulating logarithmic expressions. Key properties include the product rule (ln(a*b) = ln(a) + ln(b)), the quotient rule (ln(a/b) = ln(a) - ln(b)), and the power rule (ln(a^b) = b*ln(a)). These rules allow us to simplify and compare logarithmic equations effectively.

Natural Logarithm (ln)

The natural logarithm, denoted as ln, is the logarithm to the base e, where e is approximately 2.71828. 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 to manipulate and interpret ln expressions is crucial for solving logarithmic equations.

Equation Verification

Verifying whether an equation is true or false involves substituting values or simplifying both sides of the equation to see if they are equal. In the context of logarithmic equations, this may require applying logarithmic properties to transform one side to match the other. If the equation is false, identifying the necessary changes to make it true is a critical skill in algebra.