>> This is part twelve of
logarithms and in this video,
we solve the following
two log equations.
All right, see if you could
solve this one on your own first
by putting the video on pause.
Okay. We've got two logs
with the same base added
on the left side of the
equation, so we combine
that into one log using
the product property.
That's log base 6 of x minus
3 times x plus 2 equal to 1.
So, we have a log of
something equals one,
so now we could use
our definition;
6 to the 1st power equals
this x minus 3 times x plus 2.
Or, let's write it as x minus
3 times x plus 2 equals 6
to the 1st power.
So, now we've written
it in exponential form,
which will be a little
bit easier.
So, on the right hand side,
6 to the 1st power is just 6,
and on the left hand side; well,
we need to multiply
those two binomials.
You could do the foil method
or any method you want.
You have x squared plus
2x minus 3x minus 6.
So, we have x squared
minus x minus 6.
Now, notice I have a 6
on the right hand side.
So at this point, let's just
subtract 6 from both sides.
So, it equals 0.
Make sure you realize
one's a -6 on the left side
and there's a positive 6.
They don't cancel
each other out.
So, x squared minus
x minus 12 equals 0.
And now, we have a quadratic.
So, when you have a
quadratic equation,
hopefully you can factor it.
That's the easiest way to do it.
So, let's go ahead and factor; x
minus 4 times x plus 3 equals 0.
And, then we could set each
of these factors equal to 0.
So, we have x equals 4 or x
equals -3, and now we have
to go back to the original
problem before you did anything
to it and make sure both of
these solutions will check.
All right, so we're
ready to check.
Let's check 4 first, okay?
So, I'm just going
to [inaudible];
I'm going to plug in 4 first.
So, I have log base 6 of 4 minus
3 plus log base 6 of 4 plus 2.
So, this gives me log base
6 of 1 plus log base 6 of 6.
All right.
Now, what's the log of 1?
It's always 0, because anything,
any base that's positive
to the 0 power equals 1.
So, 6 to the 0 power
equals 1, so that's 0.
Right? And, what's
the log of 6 base 6?
Remember when these numbers
are the same, it's got to be 1,
because 6 to the 1st
power will equal 6,
so that's just the number 1.
So, you have 0 plus 1 is 1,
and that is what was
on the right side.
So, that means that 4 is
definitely a solution;
it checked.
All right, now let's do -3.
Well, what would happen if I
plugged in -3 here for the x?
There's a problem.
I'd be taking a log
of a negative number,
so without going any further,
I'm going to cross this off
because I can't take the
log of a negative number.
You don't need to go
through all of that work.
So, the solution to
this equation is 4.
That's the only one
that checked.
[ Silence ]
All right, here's the next one.
This is a little bit trickier.
Notice we've got
logs on both sides.
And, if we didn't have this
2 added to the log of x,
then we would just have; x has
to equal 3 in this case, right?
But, we have a 2.
So, we're going to have to be a
little bit more creative here.
We need to get the logs on
the same side of the equation
so we could use the product
or the quotient rule,
and it's up to you which side
of the equation you put it on.
Let's go ahead and put it
on the right hand side.
So, let's add log
x to both sides.
So, if we do that, we would
have the log of 3; I'm sorry,
I meant to subtract
log x from both sides,
so this would be minus
the log of x. Now,
also note the base is not
written, but that's okay,
because you know
what that means.
It means the base is 10.
So, we're getting closer.
We've got the log of
3 minus the log of x,
and now we can use
the quotient property.
That's the log of 3 over x. So,
remember this is the base 10,
so we said 10 to the 2nd
power equals 3 over x.
If it bothers you to see the log
on the right hand side, go ahead
and just rewrite it this way.
All right?
You just put the 2 on the right,
the other one on the left;
it's the same equation.
So, we have 3 over
x equals 10 squared.
Remember, when you don't see
the base, it means it's base 10.
[ Silence ]
Okay. Now, 10 squared is 100.
So, I have 3 over x, and
I'm going to write this
as a proportion; 100 over 1.
Because, if you write
it as a proportion,
you could say the product
of x and 100 is the same
as the product of 3 times 1.
So, 100x equals 3.
That's of course
not the only way
to solve this equation,
but that's one way.
And then, just divide by 100.
And, that's what x is.
Sorry, wish it was
easier, but it's not.
It's 3/100th's, or if you
wrote that as a decimal, .03.
So, there's more than one
way of writing that answer.
Okay, I want to go back and
redo it, putting the log
on the other side
before we check it.
So, remember we got 3/100th's,
and here's the original problem.
So, let's say you
want to put the logs
on the left side
of the equation.
Well, that's okay.
That means you're going to
subtract log 3 from both sides.
And then, you're going to have
to subtract 2 from both sides.
And so, this is what
it looks like.
So, now you have the log
of x over 3 equals -2.
And again, remembering
that the base is 10.
We'll have x over 3
equals 10 to the -2.
And then, you have to
remember what that means;
10 to the -2 means
1 over 10 squared,
which is x over 3
equals 1 over 100.
So, I hope you're seeing
what's going to happen here
when you cross multiply again.
All right?
You have a proportion.
You again are going
to get 100x equals 3,
or x equals 3/100th's, or .03.
So, it didn't matter if you put
the logs on the left hand side
and worked it, or you put it on
the right side and worked it.
Now, let's go ahead
and check it.
Okay. So, here's the
original equation.
You have to do it
in the original.
So, we have the log of;
now, it doesn't matter
if you write 3/100th's or .03.
I'm going to go ahead
and just write 3/100th's.
And, now this we
can work two ways.
You could use a calculator
by putting in; let's see.
Of course it would be
easier on a calculator.
You just put the log of .03.
So, you put that in your
calculator and you add 2
and what does that give?
Well, you're going to get an
approximation, so just keep
in mind at this point
it's not exact.
The log of .03 is -1.522
and if you round it,
it's going to be 9.
So, in other words, I'm just
going to, you know, round that.
And so, then you
have to add 2 to that
and you get approximately
.4771; that's approximate.
And then, if you then go in
your calculator and you look
up the log of 3, it has the
same approximation; .4771.
So, if you want to use your
calculator, you can get,
you know, pretty close.
But, sometimes you
might be off somewhere,
and that's because
of the rounding.
So, to check this one, it
isn't easy to do in your head.
Now, there's tricky ways to
do this without a calculator,
and that is by rewriting
2 in terms of base 10.
So, just one more thing; see
if you could follow this.
So, if you don't
have a calculator,
this is kind of tricky.
Another way of writing 2
is the log of 100, right?
Because 10 to the 2
power would equal 100.
So, if you wrote it like that,
then we could use the property
of logs over here, by
multiplying 100 times 300.
So, that says the log of 3
and then you get the
log of 3 on both sides.
So, that's a way of checking
without really using
your calculator,
in case you don't
have that on hand.
So, we did check it both ways
and our solution is .03 or,
if you want to write it in
fractional form, it's 3/100th's.
