Video Transcript
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.
