Problem 37 - Calculus with Concepts in Calculus

Question

Show that \(\ln \left(x+\sqrt{x^{2}-1}\right)=-\ln \left(x-\sqrt{x^{2}-1}\right)\) for \(x \geq 1\).

Step-by-Step Solution

Answer

The equation holds true as we verified the identity and domain conditions.

Step 1: Identify the Original Equation and Relevant Domain

You're given the expression \(\ln \left(x+\sqrt{x^{2}-1}\right)=-\ln \left(x-\sqrt{x^{2}-1}\right)\). This equation holds for \(x \geq 1\), the domain where the square root is defined and non-negative.

Step 2: Apply Logarithmic Properties

Recognize the property \(\ln(a) = -\ln(b)\) as equivalent to \(a = \frac{1}{b}\). This transforms our expression into: \[ x + \sqrt{x^2 - 1} = \frac{1}{x - \sqrt{x^2 - 1}}. \]

Step 3: Cross-Multiply to Simplify

Multiply both sides of the equation obtained in Step 2 by \(x - \sqrt{x^2 - 1}\) to eliminate the fraction:\[ (x + \sqrt{x^2 - 1})(x - \sqrt{x^2 - 1}) = 1. \]

Step 4: Simplify the Resulting Expression

Use the difference of squares formula \((a + b)(a - b) = a^2 - b^2\) to simplify the left-hand side:\[ x^2 - (\sqrt{x^2 - 1})^2 = x^2 - (x^2 - 1) = 1. \] Both sides of the equation match, confirming the identity.

Step 5: Validate the Domain Conditions

Since we used the identity \(x^2 - 1 \geq 0\) for \(x \geq 1\), all calculations remain valid as the expression inside the square root is non-negative for \(x \geq 1\).

Key Concepts

Difference of Squares

The concept of the difference of squares is a powerful algebraic identity that facilitates simplification in many algebraic expressions. It states that for any expressions \(a\) and \(b\), the expression \((a + b)(a - b)\) simplifies to \(a^2 - b^2\). This identity plays a crucial role when working with the given problem.
In our exercise, we used this identity when simplifying \((x + \sqrt{x^2 - 1})(x - \sqrt{x^2 - 1})\). The expression looks complex but employing the difference of squares makes things much simpler:
\[x^2 - (\sqrt{x^2 - 1})^2 = x^2 - (x^2 - 1) = 1.\]
This step drastically simplifies the equation and confirms the identity we aimed to prove. Remember, recognizing patterns like the difference of squares can help break down what initially seems like a complex algebraic expression.

Natural Logarithm

The natural logarithm, denoted as \(\ln(x)\), is a logarithm with the base \(e\), where \(e\) is approximately 2.71828. Unlike other logarithms, the natural log is often used in higher mathematics due to its unique properties.
In the given problem, we see an equation involving natural logarithms: \(\ln(x + \sqrt{x^2-1}) = -\ln(x - \sqrt{x^2-1})\).
One key property of logarithms helps here: \(\ln(a) = -\ln(b)\) implies that \(a = \frac{1}{b}\).
This property allowed us to transform our logarithmic equation into an algebraic equation, which could then be solved using conventional algebra techniques.

Equation Simplification

Equation simplification involves breaking down complex expressions into simpler forms, making them easier to solve. This is crucial for both solving equations and proving identities.
For this exercise, simplification steps required:

• Using logarithmic properties to transform the logarithmic equation into a fraction: \(x + \sqrt{x^2-1} = \frac{1}{x - \sqrt{x^2-1}}\).
• Cross-multiplying, which is often used to eliminate fractions and equate two expressions on opposite sides: \((x + \sqrt{x^2-1})(x - \sqrt{x^2-1}) = 1\).
• Employing the difference of squares for straightforward simplification.

Each of these steps involves recognizing opportunities to simplify and then applying the appropriate algebraic rules or identities to execute the simplification. This approach not only aids in finding solutions but also confirms the correctness of identities, as seen in this problem. Always check domain conditions to ensure all operations, such as taking square roots, are valid.