Integral Formula for Natural Log (without knowing the derivative)

This video proves that the natural log equals the integral from 1 to x of 1/t dt under the assumption that ln(x) is the inverse function to the exponential e^x. We can do this without already knowing the derivative of the natural log! More details on why the integral is the inverse of e^x: We proved in the video that any right inverse to e^x must equal that integral. However, we didn't prove that e^x has a right inverse in the first place. We know that e^x : R → R+ is a strictly increasing function whose output can be made arbitrarily large or arbitrarily small. Therefore e^x is a bijection R → R+. Every bijective function has a two-sided inverse (see [1] below). Therefore e^x has a two-sided inverse, which in particular is a right inverse. I proved in video [2] that injective functions have at most one right inverse. Therefore the right inverse to e^x is unique if it exists. But we already know that there exists one right inverse that is also a two-sided inverse. We conclude that there exists exactly one right inverse to e^x and that this right inverse is also a two-sided inverse. Hence the integral in the video is a two-sided inverse to e^x. [1] /watch/gwOrKKujn-Ejr [2]: /watch/EwD8rJuJZ5iJ8 Calculus Problems playlist: /playlist/PLug5ZIRrShJGFne7YhMi-4eYsUKzkITao Subscribe to see more new math videos! Music: OcularNebula - The Lopez

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