>> This is part eleven of
logarithms and we're going
to be solving equations
involving logs.
And, we're going to solve
the following three equations
in this video.
Some of the things you're
going to have to keep in mind
as you're doing this is
the meaning of logarithms,
how to change from a
log to exponential form,
and that a is greater than
0, b is greater than 0;
both of these numbers must be
greater than 0 and the base,
b, cannot be equal to 1.
And then, these four properties
are the most commonly used
properties when you're
solving logs.
All right, we need
to solve this.
Try it on your own first.
Okay. So, I've got the log of
2x minus 3 base 5 equals 2.
So, I could just use
the definition of logs.
What does this mean?
It means 5 to the 2nd
power equals 2x minus 3,
or 2x minus 3 equals 5 squared;
either way you want to write it.
I like to write it this way
so the variable's on the left.
And, so now this becomes a
simple equation to solve.
We just have to square 5 and
we need to add 3 to both sides.
[ Silence ]
So, we get 2x is 28.
Divide both sides
by 2, and x is 14.
Now, it's important when
you're solving log equations
that you plug the number back
in and at least make sure that,
first of all, the base is
not negative or 0 or 1,
which of course it
isn't since it's 5.
And also, what you're
taking a log of,
this 2x minus 3 also
cannot be negative.
So, when you look back
and you put in 14 for x,
you can see that this will
not be a negative number.
But, you can also
completely check it
by plugging it back
into the original.
So, we're going to check; log
base 5 of 2x minus 3 equals 2.
And, go ahead and really plug
that number in for x. So,
we have log base 5
of 2 time 14 minus 3;
let's put a parentheses
around that.
You have to make sure
you've got some parentheses.
So, we have the log
base 5 of 28 minus 3.
Well, 28 minus 3 is 25.
And, this is something you could
do in your head, because the log
of 25 base 5 means 5 to
what exponent equals 25,
and that will be 2.
Now, if it was something you
couldn't do in your head,
you can get out the
calculator and just do the log
of 25 divided by the log
of 5, and you should get 2.
And so, on the other
side we also have 2.
At the minimum, if you don't
go through an entire check,
make sure at least that
you're not ever taking a log
of a negative number.
So, for this problem,
the answer is 14.
[ Silence ]
All right.
So, try solving this
one on your own.
Notice it's not written
as a single log
on the left hand side, so
that will be your first step.
All right, so let's do that.
Where we have the log of
something plus the log
of something else, we can
use the product property.
That means it's the log
of x times x plus 21.
Now, notice that only works
if the base is the same.
And, notice you don't
see the base.
So, if there's no base shown,
remember that means
the base is 10.
So if you want, you
could write the 10.
That's optional, but
if you don't write it,
you have to at least understand
that's what is being assumed.
So, now we've got the log of
something base 10 equals 2,
so that something will
be equal to 10 squared.
So, we have x times x
plus 21 equals 10 squared.
We get that out of log form into
exponential form and now we need
to solve this equation.
Well, first I'm going to have
to distribute on the left.
And, on the right side
we need to square 10.
And now, when I analyze this,
I see that I have a
quadratic equation.
Well, to solve a
quadratic equation,
you set the equation equal to 0
and if you can factor,
you factor.
And, if you can't factor,
then you could use
the quadratic equation
or you can use the
square root property;
you know, complete the square.
But, how convenient;
this actually factors.
So, this is going to be x plus
25 times x minus 4 equals 0.
So, we set each factor
equal to 0.
So, we get two possibilities;
x can be -25 or x could be 4.
Now, remember; you must check
both of these in the original.
So, if we go back up to
the original up here,
if you plug in -25 for
x, you're taking the log
of a negative number,
and that's not allowed.
So, you could go through the
trouble of writing it down,
but hopefully right
away you'd see
that that's not going
to be a solution.
It's not that x can't
be a negative.
It's that when you're checking
it back in the original,
you can't take the log
of a negative number.
All right.
Now, if you put in 4
for x, will we be okay?
If we put in 4 here, yes,
because we're taking the log
of a positive, and if you
put in the 4 over here,
you have 4 plus 21,
also positive.
So, 4 might work, and
that's our next step is
to actually check it
completely in the original.
All right, so let's plug
in 4 for x. We have the log
of 4 plus the log of 4 plus 21.
So, that's the log of
4 plus the log of 25.
Now, there's two ways
you can go from here.
You could use your
calculator and put in the log
of 4 plus the log of
25 and add it together,
and you should get the
number 2, if you're careful.
Okay? But, if not, you
could use your property
of logs right here, and
you have the log of;
remember what this means,
you'd multiply 4 times 25.
Right? So, that would
be the log of 100.
And then, remember, even
though I didn't write it,
these are all base 10.
So, the log of 100 base 10;
well, 10 to what power is 100?
2. So, and I can see that I
have 2 on the left hand side.
I have 2 on the right hand side.
So, it checks.
So again, if you want
to use your calculator,
you could also check
it that way.
You'll put in log of 4 and then
you'll add it to log of 25.
Now, if you write down
the approximations
of your calculator, it will
not come out exactly correct,
because there's some rounding
errors that might occur.
But, if you don't round
and you do everything
in your calculator, it should
come out to 2, probably.
So, what does this tell me?
This tells me that x
equals 4 is the solution.
[ Silence ]
All right.
See if you could solve this one
on your own by putting the video
on pause and trying it first.
Okay. So again, we
don't have a single log
on the left hand side,
but we have a subtraction.
So, we can use the
quotient property.
The log base 3 of n
over 4 is equal to 2.
If you want, you
could put parentheses
around that; it's up to you.
So, the 3 to the 2nd power,
3 squared, equals n over 4
or n over 4 equals 3 squared.
So, we have n over 4 equals 9.
And, then you can just multiply
both sides by 4 to solve
for n. And, we got 36.
Now, you want to go
back up to the original
and make sure this makes sense.
So, first of all,
when you put in n,
make sure you're not taking
a log of a negative number,
and you're not, because
it will be positive there.
And now, let's do
the full check.
So, let's put in 36 for n. The
log of 36 base 3 minus the log
of 4 base 3; so, now again,
you could do this
in your calculator.
You could compute the log
of 36 base 3, which happens
to be the log of 36
over the log of 3,
or what I think would
be easier is
to simply use your
property of logs here.
The log base 3 of; that
will be 36 over 4, right?
Which is 9.
And then, what is
the log of 9 base 3;
that you could do in your head.
3 to what power would be 9?
That would be 2.
Okay? So, you have
2 on both sides,
so that means 36 is the correct
solution to that problem.
