Find log to the base three root three of 27.
Well, the first thing we’re gonna have a look at with this problem is the three root three because we want to see, can we rewrite this in different way? Can we rewrite it in a way that is three to the power of something? Well, we can rewrite the three root three. And we do that using a couple of index laws. First of all, we’re gonna start with the index law which is root 𝑥 is equal to 𝑥 to the power of a half. So therefore, we can rewrite three root three as three multiplied by three to power of a half. However, can we take it a step further? Well, we can. And that’s because 𝑥 to the power of 𝑎 multiplied by 𝑥 to the power of 𝑏 is equal to 𝑥 to the power of 𝑎 plus 𝑏.
So what we do is we add the exponents if we multiply and we’ve got the same basis. Well, three on its own is the same as three to the power of one. So we add one and a half. So we’ve got three root three is the same as three to the power of three over two. Okay, great. So now let’s put this back in to our expression. So now we have log to the base three to the power of three over two of 27. But what we do now. Well, we can also have a look at the 27. Well, because we know that 27 is equal to three cubed, we can actually rewrite this part as three to the power of something as well.
So we’ve got log to base three to the power of three over two of three cubed is our expression now. And this is really, really useful. But we ask ourselves why. It’s because of this next law, and this is one of our log laws. And what this log law tells us is that if we have log to the base 𝑎 to the power of 𝑚 of 𝑎 to the power of 𝑛, this is equal to 𝑛 over 𝑚 as long as 𝑚 is not equal to zero. Well, the key to this is that we’ve got the same value in both parts of our logarithm, and we have because we’ve got three and three. And then for our 𝑚-value, we’ve got three over two. And for our 𝑛-value, we’ve got three.
So, great. What we can do is now apply our law. But what we have now is a little bit untidy because what we’ve got is three over three over two. And what this means is three divided by three over two. Well, this is the same as three multiplied by two over three because what we do if we’re dividing by a fraction is we keep the first number the same. We change the divide to a multiply, and then we find the reciprocal of the second fraction. But now we can do is divide through by three. So now we’ve just got one multiplied by two over one, which is gonna be equal to two. So therefore, we can say that log to the base three root three of 27 is equal to two.