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. (ln x)(ln 1) = 0

검증된 단계별 안내

Step 1: Recall the property of the natural logarithm (ln) that states ln(1) = 0. This is because the natural logarithm of 1 is the exponent to which e must be raised to equal 1, and e^0 = 1.

Step 2: Substitute ln(1) = 0 into the given equation. The equation becomes (ln x)(0) = 0.

Step 3: Simplify the expression. Any number multiplied by 0 is 0, so the left-hand side simplifies to 0.

Step 4: Compare the simplified left-hand side (0) to the right-hand side (0). Since both sides are equal, the equation is true.

Step 5: Conclude that the given equation (ln x)(ln 1) = 0 is true, and no changes are necessary to make it true.

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주요 개념

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Natural Logarithm (ln)

The natural logarithm, denoted as ln, is the logarithm to the base e, where e is approximately 2.71828. It is a fundamental concept in algebra and calculus, often used to solve equations involving exponential growth or decay. Understanding the properties of logarithms, such as ln(1) = 0, is crucial for evaluating expressions involving ln.

Properties of Logarithms

Logarithmic properties are rules that simplify the manipulation of logarithmic expressions. Key properties include the product rule, quotient rule, and power rule. For instance, the property ln(a) + ln(b) = ln(ab) helps in combining logarithms, while ln(1) = 0 is essential for evaluating expressions where the logarithm of one is involved.

Evaluating Expressions

Evaluating expressions involves substituting values into mathematical formulas and simplifying them to determine their truth value. In this context, evaluating (ln x)(ln 1) requires understanding that ln(1) equals 0, which leads to the entire expression equating to 0. This concept is vital for determining the validity of the equation presented in the question.