Defining the Natural Logarithm as an Integral?!?!?

We typically define ln(x), the natural logarithm of x, by first defining the exponential function of x, noting that this is a 1:1 function, and then defining ln(x) as the inverse function to exponential. In this video we go the other way around. We define ln(x) as a particular integral, and then use that to give a new definition of e. This integral definition of logarithm has several advantages such as being able to be approximated nicely and easily proves that the logarithm function is differential. We also recover standard properties such as ln(ab) =ln(a)+ln(b). This video is part of my playlist on Calculus II:    • Calculus II (Integration Methods, Series, ...   **************************************************** Course Playlists: ►CALCULUS I:    • Calculus I (Limits, Derivative, Integrals)...   ► CALCULUS II:    • Calculus II (Integration Methods, Series, ...   ►CALCULUS III - Multivariable Calculus:    • Calculus III: Multivariable Calculus (Vect...   ►DIFFERENTIAL EQUATIONS:    • Laplace Transforms and Solving ODEs   ►DISCRETE MATH:    • Discrete Math (Full Course: Sets, Logic, P...   ►LINEAR ALGEBRA:    • Linear Algebra (Full Course)   *************************************************** ► Want to learn math effectively? Check out my "Learning Math" Series:    • 5 Tips To Make Math Practice Problems Actu...   ►Want some cool math? Check out my "Cool Math" Series:    • Cool Math Series   **************************************************** ►Follow me on Twitter:   / treforbazett   ***************************************************** This video was created by Dr. Trefor Bazett. I'm an Assistant Teaching Professor at the University of Victoria. BECOME A MEMBER: ►Join:    / @drtrefor   MATH BOOKS & MERCH I LOVE: ► My Amazon Affiliate Shop: https://www.amazon.com/shop/treforbazett