Prove an inequality bounding the difference between the natural logarithms of two numbers by the difference between the numbers themselves.

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Bounding the Difference of Logarithms using the Mean-Value Theorem

Logarithmic and exponential functions core

Prove that \(|\ln b - \ln a| < |b - a|\) for any distinct \(a, b\) in \([1, +\infty)\).

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Tests the ability to recognize when to apply the Mean-Value Theorem to prove inequalities and how to bound the derivative.

This is a classic application of the Mean-Value Theorem found in introductory calculus and real analysis courses to establish Lipschitz continuity.