In Exercises 60–63, 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. (log2 x)^4 = 4 log2 x

Verified step by step guidance

1. Identify the given equation: \((\log_2 x)^4 = 4 \log_2 x\).
2. Recognize that the equation involves logarithmic expressions and exponents.
3. Consider substituting \(y = \log_2 x\) to simplify the equation to \(y^4 = 4y\).
4. Rearrange the equation to \(y^4 - 4y = 0\) and factor it as \(y(y^3 - 4) = 0\).
5. Solve the factored equation: \(y = 0\) or \(y^3 = 4\), and check if these solutions satisfy the original equation.

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms is essential for manipulating logarithmic expressions. Key properties include the product, quotient, and power rules, which allow us to simplify or expand logarithmic terms. For instance, the power rule states that log_b(a^n) = n * log_b(a), which is crucial for transforming expressions like (log2 x)^4.

Exponential and Logarithmic Equivalence

Logarithms and exponents are inversely related, meaning that if b^y = x, then log_b(x) = y. This relationship helps in understanding how to rewrite logarithmic equations in exponential form, which can clarify whether an equation is true or false. Recognizing this equivalence is vital for solving logarithmic equations accurately.

Equation Verification

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