BAM! Mr. Tarrou In this video we're going to be solving a system of equations that
involve logarithms but first I want to say thank you to all of my students at
st. Pete I never actually stated on line where I teach but it's where than
teaching now for 24 years and you all are read the reason why I enjoy going to
work every day it's the purpose of my career and it's honestly you make my
career you're the reason why I love it and also I went to things say thank you
to everybody on YouTube that has continued to watch my videos over the
year I've really enjoyed all the positive feedback mostly and everything
all the nice comments and notes that I've gotten again through this week of
teacher appreciation this weird staying at home quarantine distance learning
thing that we are all trying to get used to and hopefully be successful and I am
struggling with this probably just as much as some of my students I may have
have nearly 700 videos on YouTube but you really need that that in-class
in-person interaction two-way communication where you can ask
questions read body language see if kids are struggling when you're saying
something that's really a challenge and I'm very proud of you for all the work
determination that you have shown over this you know situation we have to get
through so for celebration we're gonna do this really cool problem I've seen
this recently all the years have taught logarithms I've never seen a question
that involved a system of equations it's a something that I saw an exam recently
I changed the basis change the numbers made of my own and we're going to tackle
this problem what do you do when you have a system of equations log base 2 of
8 X is equal to 1 plus 2 log base 2 of Y another equation in our system is 1 plus
log base 8 of X is equal to log base 8 of 10 y minus 12 ok well first of all
don't panic it's your homework you got time to think
about it if it was on a test hopefully you've seen one of these before
your first experience on an actual test panic just kind of go back into your
toolbox what have you learned even if it's going to require skills that you
haven't necessarily used in quite a while
how do you solve an equation that has logarithms in it
well you generally need to find a way of rewriting that equation to you know say
going back into exponential form maybe that's probably the reason why the
equation has logarithms in it because in that initial form it's not necessarily
something that you can just step right into and solve you know immediately so
how are you going to do that well there's a couple of ways either we can
take this equation and write it as log base two of eight X minus 2 log base 2
of y is equal to 1 now if we do that we see we have a subtraction of two logs of
the same base and we can therefore bring this together after bringing this two up
as a power because you can't use sort of that division property for condensing
logarithms unless your coefficients are 1 so we can write this as log base 2 of
8 X minus log base 2 of Y squared is equal to 1 now we can bring this
together and write it as a single logarithm and say that we have log base
2 of 8 X minus y squared is equal to 1 and then step into exponential form and
say that we have what we have 8 x over y squared is equal to now what do you get
as an answer from a logarithm you get an exponent this is the base in exponential
form this is your exponent in exponential form and this is what the
answer is so this log base 2 of 8 x over y squared is equal to 1 can be rewritten
as 8 x over y squared is equal to 2 to the first which is just going to be 2 ok
voila now with our second equation we can look at this in a different fashion
that you can go through to write these equations in a format that do not
involve of logarithms which you can also see is
that we can think of well let's see here can we write a logarithmic expression
that has a base of eight that gives an answer of one so can I write something
like log log base 8 of something that's going to give me an answer of one well
eight to the first power remember this I'm thinking of this as being like the
answer from the logarithm eight to the first power is equal to eight so we can
write one as log base 8 of eight and actually we can also write this one
as log base 2 of 2 and that is another way of approaching this problem looking
at each of these kind of giving you a different skill or a different way of
looking at these and when we do that now we can get a piece of chalk long enough
to not scratch the chalkboard now when we look at this we can say well well
here we have log base 8 of 8 plus log base 8 of X we have a sum of logarithms
logarithms give us exponents when do you add exponents you add exponents when you
multiply like those those like bases and our like bases the base 8 of our
logarithm so we have log base 8 of ax is equal to log base 8 of 10 y minus 12 so
now that says okay if we have one term on each side of the equation and that is
involving a function of log base 8 which is a one-to-one function so the only way
that these expressions are going to be equal is if what's inside each of those
logarithms are equal so now we have 8 X is equal to 10 Y minus 12 and actually
all these terms are even so I can write this I can divide both sides by 4 write
this as 4 X is equal to 5 y minus 6 ok so now what we have is we need to take
these two equations which now we've moved out of the world
or realm of logarithms and just kind of like we're just looking at algebra now
I'm gonna go ahead and bring this Y squared over to the other side
and while I took this expression and divided everything by two just because
I'm rolling with it even though I made this example and I have this 4x is equal
to 5y minus 6 when I look at this and say 2 y squared is equal to 8x actually
this is our perfect expression that we can use for substitution and that's how
we're going to bring these together we don't have two linear functions so
combining them and this has a power of two and so power bi I think it's just
going to be neater for us to look at the system of equations with substitution so
what we're going to do now is that solving systems by substitution that you
learned back in what may be algebra 1 and do that I'm going to bring these
together by saying 8 X is the same as 10 Y minus 12 okay so now got a blue out
here differentiate these we have 2y squared is equal to 10 y minus 12 well I
guess I could've invited both sides by 2 so let's do that now we have Y squared
is equal to 5 y minus what six set this thing equal to zero so yep I made a sign
error didn't I if we take negative six and add it to
the other side it's plus six take two yes we're gonna bring that negative six
over with addition not subtraction okay now we are going to again factor this
quadratic without a copy error and we have factors of positive 6 that add to 5
we're looking at Y what minus 2 times y minus 3 is equal to 0 we're going to set
each of these factors equal to 0 and we have Y is equal to 2 and Y is equal to 3
I was pausing because I have made this example
ahead of time and I knew that there was two points that we're gonna work and why
did I pause actually why did I probably should have left that up there how did I
in my mind start to realize hey wait a minute this is looking different is it
just because I made my example well I knew there was going to be two
coordinates that were going to be solutions to this problem and whether
you do it now we're at the beginning of the problem which really probably
would've been a better idea for me whenever you are solving equations
that's when we get some red chalk out here whenever you're involved solving
for equations that have a lot of restrictions or any restrictions for
that matter on their domain you need to check that domain and make sure that you
don't end up with any extraneous solutions remember you cannot take a
logarithm of a negative value what you get from a logarithm is an exponent and
the base of that exponent is the base of that logarithmic function exponential
functions and logarithmic functions are just simply inverse math operations and
you can't have exponential functions with a negative base so you're not going
to take a positive base raise it to any kind of exponent and get a negative
answer no you can only log regardless of what the base of the logarithm is a
positive value so when you look through here we have that when you look at here
we have well okay Y has to be greater than zero well here we have the variable
of Y being used again and 10 Y minus 12 has to be greater than 0 which means
that 10 Y has to be greater than 12 not 10 12 so Y has to be greater than well
basically 1.2 we have a y-value of 2 and a y-value of 3 so both of these values
of wine are going to meet the domain restrictions for this system of
equations and just like that all right we have the X as well so we have in this
log base 2 we have an expression of 8x and we have an expression of x8 in terms
of domain is not going to make any difference we also have that X has to be
greater than zero so any solution that we come up
with for the system of equations is only a valid solution not a extraneous
solution if Y is greater than 1.2 and within each of the cordon so we come up
with X is greater than zero so both of these are going to be okay for the
domain of each of these functions well when we have a y-value of two what is
our X going to be well what's the easiest equation that relates the
variable of X and y together it seems like well I guess here we have 8x is
equal to 2y squared except now we're going to say that Y is equal to 2 so 8x
is equal to 2 squared is 4 times 2 is 8 so X is equal to 1 satisfying that and
we have one coordinate of 1 2 and now we're going to use that same
relationship between x and y with a y-coordinate of or value of 3 and we
have 8x is equal to 2 times 3 squared that's 8x is equal to 3 squared is 9 9
times 2 is 18 and X is equal to 18 over 8 or of course that can be reduced down
to be what 9 fourths so our second coordinate is equal to x
of 9 fourths and a y-value of 3 now okay good that is checking off what I got in
my pre notes by the way when I made these notes I eliminated the variable
not the variable but I eliminated the logarithmic functions both times in this
fashion where I wrote this one as log base 2 of 2 but as I started this video
I thought wait a minute I can just bring this log base 2 over with subtraction
and find another way of rewriting that equation so it does not have a
logarithmic function in it we know that both of
these points work with the domain that we identified it's also a good idea to
take these and plug of course the value of 1 & 2 into our first equation as long
long with the coordinate of 9/4 and 3 basically check each of these
coordinates in each of these equations that we'll check out so I miss a true
damn go to your homework and thank you for watching
you