Explain why the natural logarithmic function $ y = \ln x $ is used much more frequently in calculus that the other logarithmic functions $ y = \log_b x. $
The differentiation formula is simplest when $a=e$ because $\ln e=1$
So how we approach this question? We want to know why we use natural log function in calculus more than we use logarithmic function. Well, think about what calculus is all about. In calculus, we primarily use derivatives. That's one of the main cornerstones, one of the main foundations of calculus. So if we look at the derivative of the natural log of X, we see that the derivatives of D. D X of the natural log of X is equal to one over X. Okay, so one over act is a fairly simple, um, fairly simple formula for understanding the derivative. Compare that with the log rhythm of X and taking the derivative of that. And we see that that is going to be one over the natural log. One overact times the natural log of 10 is based on that alone. We see this is more complicated. Therefore, this is a simpler derivative to deal with. So since calculus requires us to use derivatives, if we're going to be taking the derivative of the natural log of X or the log of X, I would much rather take the derivative of the natural log of X because it's just gonna be something easier to deal with where, for example, if we had the natural log of X squared, it would just be one over X squared times. It's times the changeable portion two x. However, in this case it would be one over X squared natural log of 10 times two X. It just becomes much more complicated. So that's the primary reason why we use the natural log of X when using calculus.