# Logarithms explained Bob Ross style

hello and welcome to another episode of

The Joy of Mathematics I’m Toby and I’m so glad that you could

join me today I’m going to run a few examples across the bottom of the screen

of things that we will come to understand today

now I understand if you might be having a little bit of a stressful day today

because it’s not often that someone who’s studying logarithms is having the

best of days not because logarithms aren’t great,

because they are but because the process of learning to be tested can sometimes

come with a lot of stress so we’re going to start off this lesson by drawing some

trees on a sloping hill much like the one behind me today these are happy

little trees and there are lots of forests around the world filled with

trees each of them different so I want you to feel your stress melt away into

the leaves of the trees if we had a tree that was doubling in

size every year then say after four years it would be sixteen times as tall

as when we started this is not quite to scale but I think you get the idea now

mathematically we could write that down and well it is essentially two times two

times two times two is equal to sixteen you could also write that as two to the

four because there are four twos is equal to sixteen but there is one other

way that we can write it and that is log of base two 16 is equal to four so these

are equivalent this is just our logarithmic form of this up here it’s a

little bit more familiar written like this and logs are really as easy as that

students often get very intimidated by logs because the way they’re written

it’s just something we don’t have a lot of practice with but once they start to

be familiar I would hope they start to be a little easier if another tree was

to triple in size every year we can draw him getting bigger here we want to know

how many years it would take until this tree was 27 times as big as when we

started so we can write that down as three which

represents that it’s tripling this time as opposed to doubling before three to

the what and I’ll write out unknown variable as a little tree usually people

write an X here but there’s no reason it can’t be a little tree

so three to the what is going to equal 27 well just using our log rule from

before we could rewrite this as log base 3 of 27 is equal to tree our unknown

variable okay and you might be able to just figure this one out by sort of

thinking about it but in fact in this case our tree would be equal to 3. 3 to

the 3 would be 27 and that’s just another example of using this log

notation in each of these forms there is a base and exponent and an argument so

one way to remember how to go between these two things is that the base of the

logarithm is always the same as the base of the exponent in this case it is 3

here we have our exponent and here we have our argument and another way that

you could remember how to write this form out would be to draw a cross

section of a log, one that has an anti-clockwise swirl on it and you could

follow this pattern from the base to the exponent to the argument to remember the

same order to get back to this one up here let the rhythm of it flow off your

chalk like a log rolling gently down a hill. Logs will become good friends of

yours they are useful to understand really large or small numbers

but don’t get too friendly though the log function is only defined when the

base is larger than zero but not equal to one to see why if we had a log with a

base of one and say the number two well what could this equal there is nothing

that one to the power of would give you two so in fact this is undefined and it would be the

same if you had log zero of two there was no way to have zero to the power of

something giving you two also if we were to have a negative number to a

fractional power we might end up with imaginary numbers and that would be a

bit of a mess so because these choices of base are not reliable and we consider

them a little bit too flaky they’re not included as part of the function looking

back at our forest if we had a tree again that was doubling every year but

we take a look at it and it’s actually half its size then let’s think about

what’s going on here so we could write it down in Log form and what we could

say is log two, that two again represents that it was doubling, of a half because

it is now half its size is equal to what well it actually would be equal to minus

1 because 2 to the power of minus 1 would give you a half so this tree has

actually got minus 1 as I guess it’s time parameter so it’s a bit of a

time-traveling tree there but it also makes sense you know if it’s doubling

every year how far do you have to look back to see you when it was half its

size all you have to look back one year now we can do one last example with this

tree up here so say I had this written out which is log 2 of 1

well what’s going on here is that a tree is doubling in size every year and it is

just as big as when it started its value is one so how long has it been growing

for well it is actually been growing for zero amount of time you’re looking at it

right now because it is its current size just a reminder that that in our

exponential form would have been two curl around to the zero curl back to here is

equal to one there are some special logs our default spawn log is log base 10

it’s so commonly used that it’s usually just written as log such as on your

calculator we also have logs in their natural

habitat written as Ln of X these are logs in base e which is Euler’s number

and it pops up in cases of exponential growth and decay so it’s really as easy

as that so take a stroll in the woods and don’t let logs intimidate you any

longer I’d like to wish you happy studying and I hope you have an

absolutely mathematical day

The analogies that you use makes everything so much more intelligible.

Thanks pretty professor !

Unintentional math-related ASMR. It's the little things in life…

The base of my log is definitely on the uptick.

U are sooo pretty. What did you plant? Dates?

Im at LOGGerheads 😕😳

You are very accessible and clear in what you say.. A question : is your channel aimed at children ?

I had a teacher like Toby and had the same end result. I couldn't keep my mind on the lesson but I was always attentive. I remember this lesson was about trees.

DID I MISS SOMETHING IN SCHOOL

This is the most wholesome thing in the world

That's cool, but I still wouldn't have a clue how to do this without a calculator, nor do I fully understand the "why"/concept behind logs.

the beauty of maths…

i reject your premise. it would be impossible for a tree to be 16 times larger four years after planted. ergo math is impossible

Teachers like you make the world keep moving. Amazing stuff. I wish I would have had this type of tutelage back in highschool

Its a pity British schools have the dimmest, nastiest, laziest, most useless people as teachers in the whole country.

They set about making learning a form of torture.

This lady makes it effortless and fun!

Where are you, and how can I get there…you know..for some math tutoring.

Delightfully contrived, I found the video joyful, helpful and informative, thank you

I just posted this comment :- " Harshad numbers (In Sanskrit meaning 'Joy Bringers') are distinctive by being divisible by the sum total of their component integers. eg. 48, 108 & 777 are harshad numbers, 108 is sacred, and 777 is the most auspicious Angel number. so 48,108,777 is regarded in Vedic Mathematics as fairly awesome. 177 is the sum total of each line and diagonal of the smallest 3 x 3 Prime Number Magic Square and 48,108,777 divided by 177 = 271801 and 27,1801 divided by 99990 is 2.7182818281828… etc. which is not a bad approximation of Natural Log 'e' :- 2.718281828459045…etc. Just saying. " (The AI Computer must have suggested your post.)

You make me feel very stupid. Don't worry, it's not your fault, I'm just really stupid.

😍❤😍

Bob..whut?a time travelling three?a doubling tree?a spawning aspiring tree?

Tibees mam good morning 🙏…… But mam your smile and ponytail has change ….Is it burden of maths ?👌👍

take a test tube, fill it half and inverted it in pools of water, put a neodymium magnet floating inside the test tube, now we coil the test tube with copper wire so when fluctuations occurred in atmospheric pressure's then water level goes changing also magnet floating inside tube also give some emf through the coil, collected this emf in battery and use later

at gigantic level this phenomenon gives many kilowatt's of electricity forever everywhere anywhere in world😝😝😋😝

Teach me derivatives.

What a brilliant teacher , there are not many as good as you , which is more the pity !

Omg what level of simulations simulating… Oh nm

This is how math should be taught …

No idea why this was recommended to me, but after watching it, all I have to say is: "Happy studying, God bless" 🐿️

I always have a crush on my teachers

Cool, thanks for sharing.

Log lady: origins

I strangely got asmr from her explanation

As an American child of the 80's when ol' Master Sergeant Ross was on every day, it's nice to see his legacy grow. It was very enjoyable watch….And I do have a rational/irrational/imaginary number fear of Mathematics. Thanks.

mathematically when someone associated y as height of the function then it's very confusing when z direction introduced to them😱😱😱😱😈😈😈

coefficient between function and its logarithmic component😱😱😱

symbolic synonymous of mathematics symbols in literature are idiom's🐥🐥🐥🐤🐤🐤

a tree growing slowly slowly but as the new leafs are coming it's growth rate become something like exponentially so we can easily found that curve and limits of growing up of any plants

Amazing teaching style. I'm 51 and finally paid attention in math class. 🙂

Wow

Tnx today I understood why log 1 is zero in any base

Have a mathemagical day!

I'm 26 now. I never understand what Logarithm is my whole life until I see this video.

It confused me the whole time when they taught me about this in school days :))

Not so good.

You didn't cover anything different than a boring high school teacher.

Nice video but I do miss the cats!

I instantly subscribed to this lovey lady’s channel.

Thank you for making such an entertaining explanation of the limitations of logarithms.

OK, so I didn't miss the use of brackets. You obviously have a sense of humor.

I wish you were my tutor 😋

I nominate her to replace Apple’s Siri voice. 100%. All the time

@tibees Fun video! I teach math and the way I conceptualize logs I feel is much more consistent with how we teach arithmetic: If multiplication is repeated addition, and exponentiation is multiplication then division is repeated subtraction and logs are repeated division. For example: 8 divided by 2 asks, "How many times can you subtract 2 from 8 until you get to zero, the additive identity?" While log(Base2) of 8 asks, "How may times can you divide 8 by 2 until you get to one, the multiplicative identity?" For those who think of subtraction as adding until you get to a number, you can simply say for log(base 3) of 27, "How many times does 3 multiply into 27?" 3 x 3 x 3 =27, so three times. Just as, how many times does 3 add into 12? 3 + 3 + 3 + 3 = 12, so answer is 4. I do feel this is more consistent with how we teach the other operations and I'm not sure why it's not taught this way more (I have found one article that discusses logs this way). Curious what you think of this conceptual understanding of logs.

I understood that. Cool. :^)

You teach so good

Other than math

I like your smile. All win 😁👍

hotty

Came across your channel only w/in the last week or so.

Just curious about something. Under your "About" tab you mention your name is, Toby. However, your channel name, is "Tibees". Does Tibees have a special meaning?

Can you do, Logarithms explained Luigi (Mario's Plumber brother) style?

I'm curious to know what a drawing of trees on a flat hill would look like compared to trees on a sloping hill.

look, it's those infp's!

Still a complete mystery to me, but I enjoyed watching and listening to you.

I remember these stupid logarithms from high school. Big deal! What are they good for? I never understood. Log2 16 = 4. What does that prove? let me work this out. A tree doubles in height each year. How many years to reach 16 times the original height? 2x2x2x2 = 16 times the original height. So, the 2 in Log base 2 refers to the multiple that the tree height increases each year. The 16 refers to the desired multiple of the original height. How many years will it take for the tree to reach 16 times the original height? That is the 4. But what if you don't know how many years? How do you get 4 from Log2 16?

In the 2nd example, we want to know how many years it will take for the tree to grow to 27 times the original height if it triples in height each following year. Let's use H for the original height at the end of the 1st. year. Then H x 3 = H3 for the height at the end of the 2nd year. If the height triples in the 3rd year, then the height will be H3 x 3.

How do we prove this? Let's use the original height = 1 ft. at the end of the 1st. year. At end of 2nd year, the height triples to 3 ft. At end of 3rd year, the height triples to 9 ft.which is 9 times the original height after 2 years of growth from the original height of 1 ft.. At the end of the 4th year since the tree was planted, the height will be 3 x 9 = 27, or 3 x 3 x 3 = 27 ft.

Bob writes this as Log3 27 = 3.Three years AFTER the original 1st year of 1 ft. But we don't know that the answer is 3 years. How would one account to the increased growth rates? What about fractional increases? How would one arrive at fractional years in the answer?

I posted this comment yesterday. I just spent 3 hours today to revise it. I still consider logarithms as TOTAL NONSENSE!

Ha ha ha. "absolutely" she says as she brackets "mathematical" with the absolute value bars.

You are adorable!

How did you do the link to #JoyOfMathematics ? Testing #TeckBio

Ohh WOW I learned more than math by watching this channel!!! That work!

What is a "spawn log"?

Do you have a lot of views like that in Aussieand? Im impressed. I also like her video because of her voice. Aussies are cool

..where maths flies away with the fairies.

first time ever learned something while meat spanking

Can one tame a wild horsie if its grabbed firmly by the ponytail ?

Hilarious math yoga :). Loved it.

I came for her cuteness, and I ended up learning log functions.

Looks like A worlds sexiest woman teaching maths in a Beach.Awesome figures

Please, apply high frequency filter to the audio. The major part of "Bob Ross" soothing effect is the sound.

Subscribed, but not for the math

Tree fiddy

Happy Birthday, Bob!

It's all Greek to me.

When you made your variable a tiny tree, my mind was literally blown.

excuse me, sorry but where the heck is this filmed?

At about 40% i got lost in the woods

Happy little trees

Twigs

ah yes bob ross's amsr daughter… math nerd. or is it math nerd daughter… asmr?? i ferget

you are a dream

Nope, nope and nope. I still don't get it. Twice failed basic algebra, so what can you expect… WTF with the tree. I can barely rap my head around the "x"

I love your videos. Great job!

Rapunzel

W e e d e a t e r

Tibees is my youtube crush

Why use a more complicated method to arrive at the same answer? What is the benefit?

logarithms was one of my favorite topics back in high school, and i wasn’t even good at maths

I learned (and understood) more here in 8 minutes and 56 seconds than I did in all my years in school. Thank you Tobee

I never thought I could enjoy a math video

Next calc test I’m gonna use trees for my exponents

At first I thought this was a joke video, but I actually learned something.

a video about logs in a useful context!

Hello Tibees thank you for your videos, greatings from France

I wish you were my professor.😍

I'd attend every class

I hate chalk. Please use a white board.

Great lesson today! i was a big Bob Ross fan.

Came here for some asmr with a perfect subject

Yes but if a tree falls in the forest………..oh nevermind!