Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
c. ln(1 + √2) = −ln(−1 + √2)

Verified step by step guidance

1. Recall the logarithm property that states \(\ln(a) = -\ln\left(\frac{1}{a}\right)\), which means \(\ln(a) = -\ln(b)\) if and only if \(b = \frac{1}{a}\).
2. Identify the two expressions inside the logarithms: \(a = 1 + \sqrt{2}\) and \(b = -1 + \sqrt{2}\).
3. Check if \(b\) is the reciprocal of \(a\) by calculating \(\frac{1}{a} = \frac{1}{1 + \sqrt{2}}\) and compare it to \(b\).
4. Rationalize the denominator of \(\frac{1}{1 + \sqrt{2}}\) by multiplying numerator and denominator by the conjugate \(1 - \sqrt{2}\) to simplify the expression.
5. Compare the simplified form of \(\frac{1}{1 + \sqrt{2}}\) with \(-1 + \sqrt{2}\) to determine if they are equal, which will confirm whether the original statement is true or false.

Key Concepts

Properties of the Natural Logarithm Function

The natural logarithm function ln(x) is defined only for positive real numbers x > 0. It is the inverse of the exponential function e^x, and ln(a) is undefined for a ≤ 0 in the real number system. Understanding its domain is crucial when evaluating expressions involving ln.

Logarithm Identity: ln(a) = -ln(1/a)

A key logarithmic identity states that ln(a) = -ln(1/a) for positive a. This means that the negative of the logarithm of a number equals the logarithm of its reciprocal. This identity helps in rewriting and comparing logarithmic expressions.

Evaluating Expressions Involving Square Roots and Logarithms

When dealing with expressions like ln(1 + √2) and ln(-1 + √2), it is important to evaluate the numerical values inside the logarithm to ensure they are positive. Since √2 ≈ 1.414, 1 + √2 > 0 but -1 + √2 ≈ 0.414 > 0, so both arguments are positive, allowing the logarithms to be defined and compared.