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. log₄ (2x³) = 3 log₄ (2x)

Verified step by step guidance

1. Recall the logarithm property that states: $\log {\mathrm{b}}_a\left(m^n\right)=n\log {\mathrm{b}}_a\left(m\right)$. This means the exponent inside the log can be brought out as a multiplier.
2. Apply the property to the left side of the equation: $log\_4(2x^3)\; =\; log\_4(2)\; +\; log\_4(x^3)\; =\; log\_4(2)\; +\; 3\; log\_4(x)$.
3. Rewrite the right side of the equation using the distributive property: $3\; log\_4(2x)\; =\; 3\; (log\_4(2)\; +\; log\_4(x))\; =\; 3\; log\_4(2)\; +\; 3\; log\_4(x)$.
4. Compare both sides: Left side is $log\_4(2)\; +\; 3\; log\_4(x)$, right side is $3\; log\_4(2)\; +\; 3\; log\_4(x)$. Notice the coefficients of $log\_4(2)$ differ.
5. Conclude that the original equation is false because the terms involving $log\_4(2)$ are not equal. To make it true, adjust the left side to $3\; log\_4(2x)$ or the right side to $log\_4(2)\; +\; 3\; log\_4(x)$.

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, the power rule states that log_b(a^n) = n log_b(a), which is essential for comparing and rewriting logarithmic equations.

Equivalence of Logarithmic Expressions

Two logarithmic expressions are equal if and only if their arguments are equal (assuming the same base and domain restrictions). Understanding this helps determine whether an equation like log4(2x^3) = 3 log4(2x) holds true by comparing the expressions inside the logs.

Domain Restrictions in Logarithms

The argument of a logarithm must be positive. When solving or verifying logarithmic equations, it is crucial to consider domain restrictions to ensure the expressions are defined, which affects the validity of the equation and any transformations applied.