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

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Step 1: Recall the logarithmic property that states \( a \cdot \log_b(x) = \log_b(x^a) \). This property allows us to manipulate logarithmic expressions involving exponents.

Step 2: Analyze the left-hand side of the equation \((\log_2(x))^4\). This expression means that the logarithm \(\log_2(x)\) is raised to the fourth power, which is different from multiplying the logarithm by 4.

Step 3: Compare this to the right-hand side of the equation \(4 \cdot \log_2(x)\). This expression represents the logarithm \(\log_2(x)\) being multiplied by 4, not raised to the fourth power.

Step 4: Conclude that \((\log_2(x))^4 \neq 4 \cdot \log_2(x)\) because raising a logarithm to a power is not the same as multiplying it by a constant. Therefore, the given equation is false.

Step 5: To make the statement true, rewrite the left-hand side to match the right-hand side. For example, if the equation were \(4 \cdot \log_2(x) = 4 \cdot \log_2(x)\), it would be true. Alternatively, if the left-hand side were \((\log_2(x))^4\), the right-hand side would need to be \((\log_2(x))^4\) to make the equation true.

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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 solving equations involving logarithms.

Exponential and Logarithmic Equivalence

Logarithms and exponents are inversely related, meaning that if b^y = x, then log_b(x) = y. This relationship is fundamental in solving logarithmic equations, as it allows us to convert between exponential and logarithmic forms. Recognizing this equivalence helps in verifying the truth of logarithmic statements.

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 expressions. If the two sides do not match, adjustments must be made to create a true statement, which may involve applying logarithmic properties or algebraic manipulation.