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

100 thoughts on “Logarithms explained Bob Ross style

  1. 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.

  2. 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.

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

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

  5. 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!

  6. 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.)

  7. 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😝😝😋😝

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

  9. 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.

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

  11. 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

  12. 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 :))

  13. @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.

  14. 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?

  15. 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!

  16. How did you do the link to #JoyOfMathematics ? Testing #TeckBio
    Ohh WOW I learned more than math by watching this channel!!! That work!

  17. 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

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

  19. 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"

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

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