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(8x³) = 3 ln (2x)

Verified step by step guidance

1. Recall the logarithmic property that states \(\ln(a^b) = b \ln(a)\), which allows us to rewrite logarithms of powers.
2. Rewrite the left side \(\ln(8x^3)\) by expressing 8 as \$2^3\(, so it becomes \)\ln(2^3 \cdot x^3)$.
3. Use the logarithm product rule: \(\ln(ab) = \ln(a) + \ln(b)\), to separate \(\ln(2^3 \cdot x^3)\) into \(\ln(2^3) + \ln(x^3)\).
4. Apply the power rule to each term: \(\ln(2^3) = 3 \ln(2)\) and \(\ln(x^3) = 3 \ln(x)\), so the left side becomes \$3 \ln(2) + 3 \ln(x)$.
5. Rewrite the right side \$3 \ln(2x)\( using the product rule: \)3 (\ln(2) + \ln(x)) = 3 \ln(2) + 3 \ln(x)$, and compare it to the left side to determine if the equation is true.

Key Concepts

Properties of Logarithms

Logarithmic properties, such as the product, quotient, and power rules, allow us to simplify and manipulate logarithmic expressions. For example, ln(ab) = ln(a) + ln(b) and ln(a^n) = n ln(a). These rules are essential for breaking down and comparing logarithmic equations.

Equivalence of Logarithmic Expressions

Two logarithmic expressions are equal if and only if their arguments are equal, assuming the logarithm base is the same and the arguments are within the domain. This concept helps verify if an equation like ln(8x^3) = 3 ln(2x) holds true by comparing the expressions inside the logarithms.

Algebraic Manipulation of Exponents and Factors

Understanding how to factor and expand expressions with exponents is crucial. For instance, recognizing that 8x^3 = (2^3)(x^3) and that 3 ln(2x) = 3(ln 2 + ln x) helps in rewriting and comparing both sides of the equation accurately.