Video Transcript
In this lesson, we’ll learn how to
solve logarithmic equations involving logarithms with different bases. So, after this lesson, you should
be able to find a solution set from an equation containing logarithms with different
bases.
And we can see here what the base
is of a logarithm. So, we’ve got here the base and the
argument both pointed out. However, before we get on and start
to solve our logarithmic equations, there’s one thing that we want to recap. And that is how we change the base
of a logarithm. Well, let’s think about what we’ve
got here, and this is when we’re looking at changing the base of a logarithm.
So, we’ve got log to the base 𝑎 of
𝑏. But what we have is the situation
or a formula here that helps us change this to a logarithm with any base that we
want. And that is that if we have log to
the base 𝑎 of 𝑏, this can be equal to log to the base 𝑥 of 𝑏 over log to the
base 𝑥 of 𝑎, where the base can be anything we want it to be. And we can see how that’s useful in
a second cause I’ll show you an example. Well, let’s have a look at this
example to see how this becomes useful.
Well, if we’ve got log to the base
nine of 27, then we can rewrite this using the formula that we’ve got on the
left-hand side. So, this becomes log to the base
something of 27 over log to the base something of nine. And we’ve got something because
it’s up to us to decide what’s going to be useful. Well, if we take a look at 27 and
nine, we can see that, in fact, both of these are three to the power of
something. So we’ve got three to the power of
three or three squared. So therefore, we could use three as
our base because this can become very useful when we have a look at one of our log
laws in a second to actually simplify what we have here.
So what this is gonna give us is
log to the base three of three cubed over log to the base three of three
squared. And now obviously, we’ve changed
the base of our logarithm here. But what I want to do is actually
just show how we would simplify this. We do that by using our log
rules. And the log rules we’re gonna use
are that log to the base 𝑎 of 𝑚 to the power of 𝑛 equals 𝑛 log to the base 𝑎 of
𝑚. And also, log to the base 𝑎 of 𝑎
is equal to one. Well, using the first rule, we get
three log to the base three of three over two log to the base three of three.
Well, then, if you use the second
rule, we see that log to the base three of three is just going to be one. So, we got three multiplied by one
over two multiplied by one, which would give us three over two. So, what we’ve done is we’ve shown
how we changed the base of a logarithm to help us simplify cause we’ve shown that
log to the base nine of 27 would be equal to three over two.
Okay, great. So now we’ve recaped some of the
skills we’re gonna need to actually solve our equations. Let’s get on and look at some
examples of how we solve equations using the changing of base of a logarithm.
Find the solution set of log to the
base three of 𝑥 equals log to the base nine of four in the set of real numbers.
So to solve this problem, what
we’re going to do is actually change the base of our logarithm. And we do that using this formula
here, which is that log to the base 𝑎 of 𝑏 equals log to the base 𝑥 of 𝑏 over
log to the base 𝑥 of 𝑎. And in our problem, we’re only
going to change the base of the logarithm on the right-hand side of the
equation. And that’s because we’re gonna keep
it as log to the base three on the left-hand side because that’s what we’re gonna
change the base number to with our right-hand logarithm.
Well, when we do this, we get log
to the base three of 𝑥 equals log to the base three of four over log to the base
three of three squared. And what I’ve done here is actually
changed our nine into three squared because this, in fact, is the reason why we
wanted to have the base of three. And that’s because we can actually
use one of our log rules or a couple of our log rules to actually help us
simplify.
Well, the first rule we’re gonna
use is that log to the base 𝑎 of 𝑚 to the power of 𝑛 is equal to 𝑛 log to the
base 𝑎 of 𝑚. So, we’re gonna get log to the base
three of 𝑥 equals log to the base three of four over two log to the base three of
three. So now what we do is apply another
log rule, which tells us that log to the base 𝑎 of 𝑎 equals one.
And when we do that, we get log to
the base three of 𝑥 is equal to log to the base three of four over two. That’s because we had two
multiplied by one because log to the base three of three was one. So now what we can do is multiply
each side by two. And when we do that, we get two log
to the base three of 𝑥 equals log to the base three of four.
Well, now, if we actually apply our
first rule, but in reverse, we can actually rearrange the left-hand side to be log
to the base three of 𝑥 squared equals log to the base three of four. And the reason we want to do this
is cause now what we can do is actually equate our arguments because we see that
we’ve got log to the base three and then of 𝑥 squared and log to the base three of
four. So therefore, when we do that,
we’re gonna get 𝑥 squared equals four. And then when we take the square
root of this, we’re gonna get 𝑥 is equal to two.
But you might think, “Well, hold
on! What about the negative value,
because shouldn’t we have negative two or two?” Well, in fact, we’re not interested
in the negative value. And that’s because what we know is
that the argument of a logarithm must be positive and not equal to one. And as we’re looking to find 𝑥,
which is, in fact, the argument of the logarithm on the left-hand side, we can’t
have a negative value. So therefore, we can say that the
solution set of our equation is going to be two.
Okay, great. So that’s our first equation
solved. But here, we did mention
something. We mentioned that the argument of a
logarithm must be positive and not equal to one. But why is that? Well, let’s take a quick look. So, as we said, if we’ve got log to
the base 𝑎 of 𝑏, then we’d say that 𝑏 must be positive and not equal to one. But why is this? And you might actually apply some
logic and go, “Well, hold on. If log to the base 𝑎 of 𝑏 is
equal to 𝑥, then we know that 𝑎 to the power of 𝑥 is gonna be equal to 𝑏.”
Well, 𝑏 can be clearly be
negative. That’s because here’s an example
showing it. If we had negative three all cubed,
then this would give us a result of negative 27. So why can’t 𝑏 be a negative
value? Well, in fact, it’s not actually
the argument that we need to look at. It’s the base, because this is what
is our limiting factor and gives us our domain for the possible values for our
argument. Well, in fact, we know that a base
cannot be negative. And what we’re gonna do is use this
example to highlight why.
So if we had log to the base
negative two of 𝑥 equals a half, well, then what we’d have if we actually put it
into our exponent form is negative two to the power of a half equals 𝑥. So then if we wanted to solve this
for 𝑥, we’d have to take the square root of negative two. And that’s because power of a half
is the same as square root. So we’d have the square root of
negative two is equal to 𝑥, which we know is undefined.
So therefore, that shows why we
cannot have a negative base. And therefore, following from this,
if we can’t have a negative base, then we cannot have a negative argument. So that shows why it cannot be
negative. So we’re also gonna show why it
cannot be the value of one, and that’s because the base cannot be of value one. And that’s because let’s, for
instance, think if we had log to the base one of two is equal to 𝑎, then if we
rearrange this it’d be one to the power of 𝑎 equals two. Well, this can’t happen because one
to the power of anything is just one. And we’re seeing the same with the
example below.
Okay, great. So we now know why the argument of
a logarithm must be positive and not equal to one. So let’s get on and solve some more
problems.
Determine the solution set of the
equation log to the base three of 𝑥 plus log to the base 243 of 𝑥 to the power of
five plus three equals zero in the set of real numbers.
So to solve this problem, what
we’re gonna do is use our change of base formula, which we’ve got, which is log to
base 𝑎 of 𝑏 equals log to the base 𝑥 of 𝑏 over log to the base 𝑥 of 𝑎. So if we take a look at our
problem, or our equation, then what we’re gonna do is we’re gonna use this change of
base formula on log to the base 243 of 𝑥 to the power of five. And we’re gonna use it on this
because actually what we want to do is change the base to log to the base three so
we’ve got each of our logarithms with the same base.
So when we do this, what we’re
gonna get is log to the base three of 𝑥 plus log to the base three of 𝑥 to the
power of five over log to the base three of 243 plus three is equal to zero. And this is because we can decide
what the base is when we’re changing the base. And the reason this is gonna be
useful is because we know that 243 is three to the power of five. So now what we’re gonna do is use
this in a couple of our log laws to help us simplify our equation.
The laws we’re gonna look at is log
to the base 𝑎 of 𝑚 to the power of 𝑛 is equal to 𝑛 log to the base 𝑎 of 𝑚 and
log to the base 𝑎 of 𝑎 is equal to one. So when we apply these, what we’re
gonna get is log to the base three of 𝑥 is equal to five log to the base three of
𝑥, and that’s because we use our first rule for that one, and then divided by five
log to the base three of three. And that’s because, as we said, 243
was the same as three to the power of five. So then we put the five in front of
the log to the base three.
So we’ve got five log to the base
three of three, then plus three equals zero. Well, we know that log to the base
three of three is one. So therefore, we’re just gonna have
five multiplied by one on the denominator. So then what we’re gonna do is
divide through by five as well, which is gonna give us two log to the base three of
𝑥 plus three equals zero. And that’s cause we had log to the
base three of 𝑥 and then plus log to the base three of 𝑥. So that gives us two of those. So then what we’re gonna do is
subtract three from each side of the equation, which is gonna give us two log to
base three of 𝑥 equals negative three.
Well, then, what we’ve got is a
little trick to help us. And that is that we can turn the
right-hand side into something in log to the base three. And we could do that because
negative three would be the same as negative three multiplied by log to the base
three of three. So therefore, we’ve got two log to
the base three of 𝑥 equals negative three log to the base three of three. So then what we can do is apply the
converse of our first rule. So we can have log to the base
three of 𝑥 squared equals log to the base three of three to the power of negative
three.
Well, because our base is the same,
what we can do now is equate our arguments. So we get 𝑥 squared is equal to
three to the power of negative three. So therefore, 𝑥 squared is gonna
be equal to one over 27. Well, then, if we take the square
root of both sides, what we’re gonna get is 𝑥 is equal to one over root 27. We’re not interested in the
negative value because we’re told that we only want to find the set of real
numbers. And this is because 𝑥 is, in fact,
the argument in two of our logarithms, and an argument has to be positive and cannot
be equal to one.
Well, then what we’re gonna do is
simplify our root 27 by using one of our radical or surd rules. And that is that root 27 is same as
root nine multiplied by root three, which gives us three root three. So therefore, we can say that the
solution set for our equation is one over three root three.
So great, we’ve actually solved
this problem with an equation. So now, let’s take a look at one
that involves a quadratic.
Find the solution set of log to the
base two of 𝑥 equals log to the base four of three 𝑥 plus 28 in the set of real
numbers.
So in this problem, what we can do
is use the change of base formula. So log to the base 𝑎 of 𝑏 equals
log to the base 𝑥 of 𝑏 over log to the base 𝑥 of 𝑎. And we’re gonna use it because what
we want to do is have the right-hand side with same base as the left-hand side. So when we do that, what we’re
gonna get is log to the base two of 𝑥 equals log to the base two of three 𝑥 plus
28 over log to the base two of four.
And whenever we’re solving a
problem like this, what we do look out for is where we could actually have something
where we’ve got, for instance, log to the base two of two. So it’s the same as the form log to
the base 𝑎 of 𝑎. And in fact, we can have that here
because four is the same as two squared. So we can rewrite it as our
denominator, being log to the base two of two squared.
So now, to simplify further, what
can do is apply a couple of our log rules. First is that log to the base 𝑎 of
𝑚 to the power of 𝑛 is equal to 𝑛 log to base 𝑎 of 𝑚. And then, we’ve got log to base 𝑎
of 𝑎 equals one. So applying the first rule, we’ve
got log to the base two of 𝑥 equals log to the base two of three 𝑥 plus 28 over
two log to the base two of two. And then applying the second rule,
we can say that the denominator becomes just two because it’s two multiplied by one,
as log to the base two of two is just one.
So then, if we multiply both sides
by two, we get two log to the base two of 𝑥 equals log to the base two of three 𝑥
plus 28. So now what we’re gonna do is take
a look at the left-hand side and apply the converse of the first rule we looked at
for our logarithms. And when we do that, we get log to
the base two of 𝑥 squared equals log to the base two of three 𝑥 plus 28. So now what we can do is actually
equate our arguments because we have the log to the same base on either side of the
equation.
So when we do that, we now have 𝑥
squared equals three 𝑥 plus 28. So now what we can do is rearrange
to actually change this into a quadratic that’s equal to zero. So we’re gonna subtract three 𝑥
and subtract 28 from each side. So, when we do that, we get the
quadratic 𝑥 squared minus three 𝑥 minus 28 equals zero. So now, what we need to do is solve
this for 𝑥. So, if we want to solve 𝑥 squared
minus three 𝑥 minus 28 equals zero, what we’re gonna do is factor it. So, if we factor this quadratic,
we’re gonna get 𝑥 minus seven multiplied by 𝑥 plus four equals zero. So therefore, we get 𝑥 is equal to
seven or negative four.
So our solution set then is gonna
be seven or negative four, isn’t it? Well, no because 𝑥 cannot be one
of our values. The value it can’t be is negative
four. And that’s because we know that if
we take a look back at our equation, 𝑥 is the argument of the left-hand side. And we know that the argument must
be positive and not equal to one. So therefore, the solution set of
our equation is just seven.
Okay, so there we solved a problem
with quadratic. What we’re gonna do now is move on
to a problem where, in fact, we’re gonna have more than one result in our set.
Solve log to the base two of log to
the base three of 𝑥 squared minus eight 𝑥 equals one where 𝑥 is in the set of
real numbers.
So the first thing we can do with
this problem is make sure that we’ve got a log to the same base on each side of the
equation. And we can do that by applying one
of our log rules. And that is that log to the base 𝑎
of 𝑎 equals one. So, therefore, we can say that log
to the base two of log to the base three of 𝑥 squared minus eight 𝑥 equals log to
base two of two because, as we said, one is just the same as log to the base two of
two.
So now the reason we’ve done this
is because what we’ve got is log to the same base on the left- and right-hand
sides. So, what we can do is equate our
arguments. So, we can say that log to the base
three of 𝑥 squared minus eight 𝑥 is equal to two. So now, to solve for 𝑥, what we
can do first is actually rearrange from logarithmic to exponent form. So if we have log to the base 𝑎 of
𝑏 equals 𝑥, then we say that 𝑎 to the power of 𝑥 is gonna be equal to 𝑏.
So therefore, if we identify our
𝑎, 𝑏, and 𝑥, well then we can rewrite our equation as three squared equals 𝑥
squared minus eight 𝑥, which is gonna give us nine equals 𝑥 squared minus eight
𝑥. So then, what we’re gonna do is
subtract nine from each side of the equation. So, we get zero equals 𝑥 squared
minus eight 𝑥 minus nine. So now what we need to do is solve
this quadratic.
Well, we can solve the quadratic by
factoring. And if we do, we get zero equals 𝑥
minus nine multiplied by 𝑥 plus one. So therefore, we can say that 𝑥 is
gonna be equal to nine or negative one. So great, and these both are going
to be in our solution set. Well, you might think, “Well, no,
it can’t be because we know that the argument must be positive and not equal to
one.” So therefore, we cannot have the
negative value. However, that is not the case in
this particular problem because if we take a look at the argument, the argument is
not 𝑥. The argument is 𝑥 squared minus
eight 𝑥, well, the argument of one of our logarithms.
And what we’re gonna do to show
that is actually substitute in 𝑥 equals negative one into this argument. Well, if we do that, we get
negative one all squared minus eight multiplied by negative one, which is gonna give
us one plus eight because negative one squared is just one. And if you subtract a negative,
it’s the same as adding a positive. Well, this gives us the value of
nine, which is both positive and not equal to one. So therefore, it satisfies the
conditions that we have for our argument. So therefore, we can say that the
solution set for our equation is nine and negative one.
Okay, great. We’ve looked at a range of
different problems. So now, let’s take a look at a
summary of the key points. So, if we take a look at our key
points, the first one is that if we want to change the base of one of our
logarithms, then we can say that the log to the base 𝑎 of 𝑏 is equal to the log to
the base 𝑥 of 𝑏 over the log to the base 𝑥 of 𝑎. So, we can actually have the base
changed to anything that we want. And what we tend to do is choose a
base that’s gonna be the most useful when we’re looking to solve an equation or
simplify an expression.
Then what we also looked at is that
if we have log to the base 𝑎 of 𝑏, then the 𝑏 would be our argument and the 𝑎
would be our base. And the argument must be positive
and equal to one. And in fact, this is because the
base must also be a positive value, not equal to one or zero. And this is important because we
need to use this when considering the domain of 𝑥 if 𝑥 is part of an argument of a
logarithm in an equation that we’re trying to solve.
