Question Video: Finding the Solution Set of Logarithmic Equations over the Set of Real Numbers Mathematics

Find the solution set of log₉ 216 × log₉ 81 = log₉ 36 × log₉ 𝑥⁶ in ℝ.

04:45

Video Transcript

Find the solution set of log to the base nine of 216 multiplied by log to the base nine of 81 equals log to base nine of 36 multiplied by log to base nine of 𝑥 to the power of six in the set of real numbers.

So the first thing we can do in this question is have a look at the numbers we’ve got involved and see if we can actually rewrite them in a different way. So, first of all, we have 216. Well, we know that this is six to the power of three. Then, next, we have 81, and this is nine squared. Then, if we move along, we have 36, which is six squared. And then for the final term, we have 𝑥 to the power of six. And this is 𝑥 that we’re trying to find because we want to find the solution set. Okay, so let’s rewrite our equation.

And when we do that, what we have is log to the base nine of six cubed multiplied by log to the base nine of nine squared equals log to the base nine of six squared multiplied by log to the base nine of 𝑥 to the power of six. So now, we can rewrite our equation once more. And we can do that using one of our logarithm rules. And the first one we’re going to use is the power rule, which tells us that log to the base 𝑏 of 𝑥 to the power of 𝑝 is equal to 𝑝 multiplied by log to the base 𝑏 of 𝑥. And when we do that, what we have is three log to the base nine of six multiplied by two log to the base nine of nine equals two log to the base nine of six multiplied by six log to the base nine of 𝑥.

And now, if we take a look at the second term, what we can do is apply another relationship we know about logarithms. And that is the log of one, which tells us that log to the base 𝑏 of 𝑏 is equal to one. Well, we’ve got here two log to the base nine of nine, which is going to be two multiplied by one, because log to the base nine of nine is just one. So it’s gonna give us two. Well, therefore, if we’ve got three log to the base nine of six multiplied by two, it’s gonna give us, on the left-hand side, six log to the base nine of six. And then this is equal to two log to the base nine of six multiplied by six log to the base nine of 𝑥.

Well, now, if we inspect a little further, we can see that we’ve got log to the base nine of six on the left-hand side and log to the base nine of six on the right-hand side. On the left-hand side, we have six log to the base nine of six. And on the right-hand side, we have two log to the base nine of six. So therefore, this tells us that six log to the base nine of 𝑥 must be equal to three. And that’s because three multiplied by two log to the base nine of six would give us our six log to the base nine of six.

So, as we said, therefore, six log to the base nine of 𝑥 is equal to three. And then what we can do is divide through both sides by six. And once we’ve done that, what we get is log to the base nine of 𝑥 is equal to one-half. So now, so we can solve to find 𝑥, what we do is consider the general form of the logarithmic equations. And that is that if we have log to the base 𝑏 of 𝑥 equals 𝑘, then 𝑥 is equal to 𝑏 to the power of 𝑘. Well, applying that, we can see that our 𝑏 is nine, our 𝑥 is 𝑥, and our 𝑘 is a half. So therefore, we can say that 𝑥 is gonna be equal to nine to the power of a half. And we can rewrite this as 𝑥 is equal to root nine. So therefore, we can say that 𝑥 is gonna be equal to three.

You might think, “Oh, hold on. Actually, root nine will give us two solutions, positive or negative three.” However, we’re not interested in negative three. And that’s because when we’re looking at logarithms, the argument can never be a negative value. And that’s because logarithms of negative numbers are not defined in the real numbers. Okay, great. So we know that 𝑥 is gonna be equal to three. So therefore, we can say that the solution set of log to the base nine of 216 multiplied by log to the base nine of 81 equals log to the base nine of 36 multiplied by log to the base nine of 𝑥 to the power of six in the set of real numbers is three.

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