Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
c. ln xy = (ln x)(ln y)

검증된 단계별 안내

Recall the logarithm property for the product of two positive numbers: \(\ln(xy) = \ln x + \ln y\). This is a fundamental identity in logarithms.

Compare the given statement \(\ln xy = (\ln x)(\ln y)\) with the known property. The statement suggests that the logarithm of a product equals the product of the logarithms, which differs from the sum in the known property.

To test the statement, consider specific positive values for \(x\) and \(y\), for example, \(x=2\) and \(y=3\). Calculate both sides: \(\ln(2 \times 3)\) and \((\ln 2)(\ln 3)\) to see if they are equal.

Since \(\ln(6)\) is approximately \(1.79\), and \((\ln 2)(\ln 3)\) is approximately \(0.48\), the two sides are not equal, providing a counterexample.

Conclude that the statement \(\ln xy = (\ln x)(\ln y)\) is false because the logarithm of a product is the sum of the logarithms, not their product.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Properties of Logarithms

Logarithms have specific properties that simplify expressions, such as ln(xy) = ln(x) + ln(y). Understanding these properties helps determine the validity of logarithmic equations.

Difference Between Addition and Multiplication

In logarithmic identities, multiplication inside the log translates to addition outside, not multiplication. Recognizing this distinction is crucial to avoid incorrect assumptions like ln(xy) = (ln x)(ln y).

Counterexamples in Mathematical Proofs

A single counterexample disproves a general statement. Testing the equation with specific values of x and y can show whether ln(xy) = (ln x)(ln y) holds true or not.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.c. ln xy = (ln x)(ln y)