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