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)

Verified step by step guidance

1. Recall the logarithm property for the product of two positive numbers: \(\ln(xy) = \ln x + \ln y\). This is a fundamental identity in logarithms.
2. 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.
3. 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.
4. 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.
5. 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.

Key Concepts

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.