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Video: Finding the Solution Set of a Logarithmic Equation over the Set of Real Numbers

Alex Cutbill

Find the solution set of log₃ 𝑥 = log₉ 4 in ℝ.

03:22

Video Transcript

Find the solution set of log base three of 𝑥 equals log base nine of four in the set of real numbers.

Obviously, we would like to find the value, or possible values of 𝑥, and we have log base three of 𝑥 on the left-hand side. How can we turn that into 𝑥? Well, three to the power of the left-hand side will be three to the power of the right-hand side, and of course three to the power of log base three of 𝑥 is just 𝑥. This is just a special case of 𝑏 to the power of log base 𝑏 of 𝑥 equals 𝑥 with 𝑏 equal to three.

So we have that 𝑥 is equal to three to the power of log base nine of four. Can we simplify this right-hand side? Well, yes. Actually, if you put it into a calculator, you’ll find that three to the power of log base nine of four is just two. Can we show this without using our calculator? Well, yes. Three to the power of log base nine of four times three to the power of log base nine of four is nine to the power of log base nine of four. This is just a special case of 𝑎 to the power of 𝑛 times 𝑏 to the power of 𝑛 being 𝑎 times 𝑏 to the power of 𝑛. Here both 𝑎 and 𝑏 are three, and 𝑛 is log base nine of four. And nine to the power of log base nine of four is four.

So if we take square root on both sides, we see that three to the power of log base nine of four is either plus or minus two. And of course when you raise a positive number to the power of some other number, you always get a positive number. So three to the power of log base nine of four must be equal to two. So we can write down 𝑥 equals two. But of course we’re looking for the solution set of this equation, so we need to put that solution, two, into a set.

The solution set is the set containing the number two and nothing else. Instead of trying to get 𝑥 on the left-hand side straightaway, we could’ve decided to make the right-hand side be log base three of something. We could then divide both sides by log base nine of three. We chose to do this because we can then apply the change of base formula: log base 𝑘 of 𝑎 over log base 𝑘 of 𝑏 equals log 𝑏 of 𝑎, where 𝑘 is nine, 𝑎 is four, and 𝑏 is three. And so we get log base three of something on the right-hand side.

What is log base nine of three? Another way of asking that, is to say nine to the power of what is three? The answer is: a half. Therefore, the left-hand side is log base three of 𝑥 over a half. Multiplying both sides by a half, we get log base three of 𝑥 is a half log base three of four.

And it turns out that we can simplify the right-hand side using another law of logarithms. The right-hand side is log base three of four to the power of a half, which is log base three of two. And so just like before, we get 𝑥 equals two.