Defining the natural logarithm

The definition of the natural log as an area accumulation function. #shorts #math #manim #logarithms #calculus #integral

[OSA]

This thing suddenly started to make sense.

[prabhatpramanik]

always wanted a maths teacher like him

[pengin]

Here's something interesting. People learn how to integrate x^n in school, the antiderivative is 1/(n+1) x^(n+1) +C and they also learn that you cannot do this for n=-1, because then you divide by 0. However, one might feel like it just *should* somehow work. Well, what happens when we try to integrate x^(-1+eps) for some small eps>0? Well, the antiderivative is 1/eps * x^eps + C. If we choose C smart in such a way that for x=1, we get that this antiderivative is 0, we get C= -1/eps, so our antiderivative is (x^eps - 1)/eps. And if we now let eps go to 0, the limit of this is precisely ln(x)

[bjchoyce]

I really love these videos man. It is really cool to see these visual representations of sums, series, and other graphs! Keep it up!

[TacticusPrime]

I learned calculus after logarithms and I never really went back to them. Natural log never clicked in my head.

[davidphy]

Thank you very much indeed, beautifull explanation.

[hugoviloriagonzales]

Saludos desde México, éstas modelaciones, amigablemente nos anticipan y preparan, para facilitar el interés de entender, como representarlo con sus correspondientes mecanismos simbólicos.

[thebread]

I love these videos please keep uploading these simulations:)

[alute]

these were the most valuable moments of my life. thank you for sharing

[wsar]

omg I've always struggled with the idea of the natural log, what it is and where it comes from, and now I understand it! Thank you so much

[camnewton]

I can’t be the only person like just out of college who wishes they watched more of these shorts back in college lol

[goodguyamr]

i don't know anything about natural logs but I still watched to the end

[skatheo]

i'd love to see the visual correlation with log (as the potence you'd have to raise e to reach x) and 1/x

[alvargd]

integral tip: you can do \int f(x)\, dx instead of just \int f(x)dx because \, leaves a nice space inbetween

[AnglandAlamehnaSwedish]

Man ty for ur videos i hope my kid gets to see these, these i wish were around like if sal khan was around or Google was around wen i was a maths major in the university

[franklinmontez]

what a great way to show natural logarithm, i’ve had a hard time visualizing it until now

[gabedarrett]

There's another derivation using implicit differentiation:
y=ln(x)
e^y=x
differentiate both sides with respect to x
e^y × dy/dx = 1
dy/dx = 1 ÷ e^y
But recall that e^y = x from the beginning. Substituting yields...
dy/dx = 1/x
dy = 1/x dx
integrate both sides
y = integral of 1/x dx
recall that y = ln(x) from the beginning.
ln(x) = integral of 1/x dx
And that's why the antiderivative of 1/x is ln(x)

[angelmendez-rivera]

I see people in the comments saying that this definition is unintuitive and unnatural, because it fails to explain the relationship between the functions log, exp, and the real number e. This is incorrect, and I will explain why here.

The definition is natural, because the most basic type of function everyone is familiar with is a rational function. As such, finding the cumulative integrals of rational functions is something we would naturally ask ourselves about, and want to investigate, almost immediately as soon as we are introduced to the concept of the integral and its definition. The inevitable question you will stumble upon is, "what is this cumulative integral of the reciprocal function"? This is especially so, as among the rational functions, the reciprocal function is incredibly basic, and incredibly important: it is one the first functions you learn about when studying mathematics in primary school. The reason this makes for a natural avenue to define the logarithm is because, merely by asking the question, you have already defined the function!

All that remains is to study its properties: in particular, to prove that from this definition, (a) for all real x, y > 0, log(x·y) = log(x) + log(y), (b) the inverse of log is differentiable, and it satisfies the equation y' = y. Michael Penn (the YouTube channel) has a very recent video in which he goes over proofs of these, but there are many other videos on the Internet that cover this topic as well. As such, there is a very natural and easy-to-derive relationship between exp and log when defined in this manner. As for the relationship between log and e, this can be derived by proving from exp' = exp that for all x, y, exp(x + y) = exp(x)·exp(y), by using the binomial theorem and the Maclaurin series expansion, and once you have this property, you can note that exp'(x) = lim (exp(x + k) – exp(x))/k (k —> 0) = exp(x)·lim (exp(k) – 1)/k (k —> 0), which implies that lim (exp(k) – 1)/k (k —> 0) = 1. Using some substitutions, one can easily prove that lim (exp(k) – 1)/k (k —> 0) = 1/ln(lim (1 + 1/m)^m (m —> ∞)), and since this must be equal to 1, one has that ln(lim (1 + 1/m)^m (m —> ∞)) = 1, which means exp(1) = lim (1 + 1/m)^m (m —> ∞), and we simply call this quantity e, so that exp(1) = e, and ln(e) = 1. Finally, for a bonus, you can then use the binomial theorem again and the Maclaurin series expansion of exp to prove that exp(x) = lim (1 + x/m)^m (m —> ∞).

This works out much more naturally than other definitions of log, especially since trying to define exp on its own merits, before defining log in terms of exp, does not come up anywhere nearly as inevitably and naturally.

[Champtiago]

I don’t even know what a logarithm or integral is but I still liked your video

[quantum_psi]

I don't understand the integration intuitively because if you look at the limit as x approaches positive infinity for f(x), the area gets smaller. How can an area that decreases over time equal the natural log which increases in the area?