Logarithm Fundamentals | Ep. 6 Lockdown live math

Back to the basics with logarithms.

Triangle of power:

Beautiful pictorial summary by @ThuyNganVu:

Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld

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Music by Vincent Rubinetti.
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Various social media stuffs:

[carlos]

(8:55) Intuition for Logarithm
(14:11) Triangle of Power / Relationship between Exponentiation, Logarithms, Roots
(20:11) Product Identity
(24:35) Power Identity
(31:54) Reciprocal
(43:44) Can you have log base zero?
(45:29) Richter Scale
(55:46) log base 2 and base 10
(01:04:50) Change of Base
(01:15:09) Change of Base 2
(01:23:15) Challenge

One of the best ways to get intuition is just plugging in examples

[kiiometric]

When you say that the ones that never learnt logarithms shouldn't be intimidated by the ones who had, it's so wholesome and inspiring, you are by far the best teacher I've seen and watched

[gayatrisavarkar]

In the last question, if you use the other formula, it becomes really quick : logb(a) = 1/ loga(b), so we immediately get the whole expression as log100!(2) + log100!(3) + .... + log100!(100)
Then we use the log(a) + log (b) = log(a.b) and rest follows. Using change of base formula takes more time, this method is quicker.

[sailor]

Imagine you walk into into a classroom.
You see a few kids in the front of the class and in a back of the class a bunch of bearded men in suits being like: "Don't mind us, we are just taking notes".

[ajhcornwall]

Awesome lesson. You’re the best maths teacher I’ve come across. Clear, precise, entertaining, fun, enlightening. Keep up the good work!

[TheNethIafin]

In Ukraine we were taught like this:
ln - log base e
lg - log base 10
lb - log base 2
log - you have to specify base

also tg - tan, arctg - arctan, and other stuff I sometimes can't get used to when I watch lectures in other languages

[notabotta]

Grant: “this series is directed towards high school students”

Also Grant: “if you have a child around the age of 5...”

[CTK]

After Grant told me that log(a+b) didn't have a nice and clean formula for it, I had to do some exploration to see if I could find anything myself, and I have! Mind you, I'm paused at 52:29 so if he already said this later in the video, then my bad for commenting too early. log(a+b) is a pretty weird one to think about, but try to imagine each stretch from the powers of the base being proportionate. For example, just think about the graph of f(x) = log(x) where the base can be anything. Let's say the base is n. Look at the graph from x=1 to x=n. I'm looking at 1 to n because this translates to x=n^0 and x=n^1. If you look how the graph grows from x=n^0 to x=n^1, you'll notice that the graph grows at the same rate between x=n^1 and x=n^2. You'll notice that the y-value's decimal places look very similar at midpoints. You begin to notice that each section of this graph from x=n^(k) to x=n^(k+1) look exactly the same, and that's because it's is (at least proportionately)


So how does this help, intuitively? Let's look at an example of log(8+16) using a base of 2. We know that log(8) = 3, and log(16) = 4. But what about this pesky log(24)? 24 isn't a nice power of 2 so this feels pretty weird, but it just so happens that 24 is the midpoint between 2^4 and 2^5. So the answer of log(24) is going to be between 4 and 5, and we know that the graph of f(x)=log(x) between x=16 and x=32 is proportional in growth rate to that same graph between x=1 and x=2 (because that's x=2^0 and x=2^1). So all we have to do is figure out how deep into the section between x=16 and x=32 we went. We went half way, to x=24. So what's the value half way between x=1 and x=2? Well it's going to be the y-value at x=1.5 because that's half way. Another way of writing that is the y-value of (1+b/a), which is (1 + 8/16), which is 1.5. Take that log.. log(1.5)... or log(1+a/b)... and add it to log(16), and that's your answer. Yes! log(8+16) = log(16) + log(1 + 8/16). I should also note that you can write it the other way around: log(8+16) = log(8) + log(1 + 16/8).


Generally, log(a+b) = log(b) + log(1 + a/b)
AND you can even swap the (a) and (b) on the right side: log(a+b) = log(a) + log(1+ b/a)

[NikolajLepka]

17:20 One could argue that the radical notation shouldn't even be used at all, since roots are nothing more than some number raised to a fraction

[junkmail]

25:50 You can see the people working out the details. There was a sudden flash of one answer, but the other 3 were in the ones, sort of even, for a while. Then there was a build on the lower three, but a real powerful build on the leading answer, as if people were finding reasons to exclude certain suspected answers, while the strong build on the leader, people were checking and confirming their strong suspect (the flash). I found this interesting. Determining the calculation time of different paths.

[lawrenceora]

Thank you so much Grant. I'm about to finish my degree in computer science and I just realized log base 2 is also the number of 0's in the binary representation.

[squibble]

the triangle thingy is what helped me remember what a logarithm is in the first place :D

[matron]

1:30:46 best moment of the stream haha. „41 of you want me to come up with a number fact numerical 69, but I won’t“ This really made me laugh😂 Thank you for making such cool streams Grant!

[kenhaley]

I absolutely LOVE the triangle notation!

[masukki]

I gave my students a link to this live stream this morning. I really hope that they watched this, because next week I might use all of your questions as a part of my test.

[itchy]

I like thinking that logarithms lower the complexity of the input: (exponents -> multiplication -> addition)

It really helped me get through math class, but I don't know if that's the "right" way to look at them.

[notabotta]

1:07:52 is how I always remembered the change of base formula. Thank you for being the first one to actually help me understand why! I don’t think any of my teachers understood the intuition behind it.

[mipsuperk]

40:10 I have access to small children. Here are the answers I got: (age 4) "6", (age 6) "8", (age 9) "4.5? Wait no, that's halfway between 0 and 9. I don't know. The question doesn't make sense." (Edited time).

[mavericksantiago]

A great vedio ... when I was taught logarithms our teacher used to say if you want to compute fast then I need to learn it . So we also had a printed log tables to the base 10. Any number can be broken down to base 10 to power and then the remaining number can be added . That time during the early 90's we were not allowed to use calculators in Indian school... this generated my intrest in logarithms... but once I reached high school and then engineering we never bothered about logarithms other than using it in graphs....

[garyb]

Thank you for teaching me logarithms. Many concepts that you covered had caused me confusion in the past.
You tell you how well you taught me, I was able to work out the final question in my head, although I converted everything to log base 100! and got the same pattern.