Uploaded January 2023 | Updated September 2026, 2 weeks ago
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] youtu.be/E-njuKKrOwg
[2]: youtu.be/i5ZJuJr8DwE
Calculus Problems playlist: youtube.com/playlist?list=PLug5ZIRrShJGFne7YhMi-4eYsUKzkITao
Subscribe to see more new math videos!
Music: OcularNebula - The Lopez
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] youtu.be/E-njuKKrOwg
[2]: youtu.be/i5ZJuJr8DwE
Calculus Problems playlist: youtube.com/playlist?list=PLug5ZIRrShJGFne7YhMi-4eYsUKzkITao
Subscribe to see more new math videos!
Music: OcularNebula - The Lopez
