hi class this video is on logarithms logarithms
sound complicated but they are basically the
inverse of exponential functions they just
extract the exponent they are actually quite
simple so let's get started okay so what is
the log base 10 of 1000 well it's equal to
three and let's check that on the calculator
okay if i put in log of 1000 i get out three
now why is that well basically because it's
kind of like moving this ten over here 1000
is equal to 10 cubed so when i take the log
by the way log base 10 is the same as log
on your calculator it just assumes base 10
log base 10 of 1000 well i'm asking what number
is 10 raised to to equal 1000 and that number
is three if we have log base two of 16 that's
equal to well it's asking for sixteen is equal
to two to what number and that number is four
and so log base two of 16 is four log base
10 of 10,000 is equal to what well we're asking
is 10,000 it's equal to 10 to what number
and it's four so that number is four so we're
really just extracting the 10 exponent out
of this number we're extracting the two exponent
out of this number log base three of nine
is just equal to two because i'm taking the
three exponent out of the nine nine is equal
to three squared and so i'm just extracting
that exponent 
okay so in general if we have log base a of
x and that's equal to some number b then i
just took the a exponent out of x and got
b that meant that x was equal to a to the
b okay i'm taking the a exponent out of x
and that's what it means that's a logarithmic
form this is kind of the exponential form
there they're inverses of each other really
okay so what's log base 10 of 10 to the seventh
well what is the 10 exponent of 10 to the
seven it's seven so that's what it's equal
to log base two of 32 or two to the fifth
is just equal to five i'm extracting the two
exponent out of this and it's clearly five
log base two of two to the x therefore is
just equal to x what's the exponent of two
to the x well it's x and right there you notice
from our previous video on inverse functions
we have one function f(x) is equal to the
log base two of x and we have another function
inverse of x or g of x is equal to two to
the x and they're inverses of each other because
f of f inverse of x is going to be equal to
log base two of two to the x what's the two
exponent of two to the x well it's just equal
to x okay remember log base three of 81 what's
that equal to well 81 is three to the fourth
log base three of three to the fourth we're
extracting the three exponent out of that
so it's just equal to four okay a way to remember
the exponential form 
and this is the log form 
this is really the definition is we can really
just kind of move this a over to there and
we get x is equal to a to the b okay well
let's see a graph of these let's go to our
desmos calculator and i put these in already
and we have the function y is equal to two
to the x and that's equal to this function
that we saw before and the line y is equal
to x and we have the log base two of x and
it looks like this those are inverse functions
okay whenever we have a point here okay that's
a point zero one and then we have the corresponding
point over here one zero okay y interchanges
with x this one brings it to some y and that
takes y back to x so if we have the point
right here two four we're going to have another
point here four two okay it takes y back to
x and here we should have the point three
eight and we'll have the point eight three
over here okay so every time there is an x
y over here there is a y x over here and so
it just flips it about the line y is equal
to x okay just flips it about the y y is equal
to x and that's what an inverse function looks
like okay so if we have the function f(x)
is equal to log base two of x then we put
some values in for x if x is equal to two
then what do we have well that's log base
two of two well that's the same as two to
the one so the exponent is one so we get one
okay now what about log what about one here
if x is equal to one what's log base two of
one well that's also equal to log base two
of two to the zero right two to the zero is
one so that's equal to zero so that points
on every log function okay and now what about
one half what's log base two of one half well
one half is equal to two to the minus one
so there's our exponent just extracting the
exponent is equal to minus one so we get one
half for x we get minus one here so one half
minus one and what about log base two of eight
well that's equal to log base two of two cubed
so i'm extracting the two exponent so it's
three so i get eight and then point three
so i got eight and three here what about four
i get two right because log base two of four
four is two squared so i get two and i get
two here and this function looks like this
the inverse of my two to the x so here's a
log function on base two of x okay and what's
the domain here and range okay well the range
goes from minus infinity all the way to infinity
but the domain is i can only put in positive
values for x i can't put in a negative number
two to the x it looks like this it's never
negative so i can't get back there okay so
the domain is zero comma infinity and note
that this is the range of the inverse of two
to the x and the domain for two the x my domain
is from minus infinity to infinity but my
range is zero to infinity and they flip for
this one and that's basically because every
time i have a point on this graph okay i don't
have it drawn well but every time there's
an x y over here there's a corresponding y
x over here on the inverse function okay alright
so what did we do for 
logarithmic things we got the graph of a log
it's the inverse of two to the x and it just
extracts the exponent okay so i hope you found
this helpful introduction to logarithms video
useful and it helped you in your learning
and that you enjoyed it cheers
